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As of 3.4.0b1, the following test failures are observed with Tk 8.5 on OS X (python.org 64-/32- installer with ActiveTcl 8.5.15.1 on OS X 10.9):
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FAIL: test_bitmap (tkinter.test.test_tkinter.test_widgets.ButtonTest)
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Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 240, in test_bitmap
 errmsg='bitmap "spam" not defined')
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 75, in checkInvalidParam
 widget[name] = value
AssertionError: TclError not raised
======================================================================
FAIL: test_bitmap (tkinter.test.test_tkinter.test_widgets.CheckbuttonTest)
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Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 240, in test_bitmap
 errmsg='bitmap "spam" not defined')
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 75, in checkInvalidParam
 widget[name] = value
AssertionError: TclError not raised
======================================================================
FAIL: test_insertborderwidth (tkinter.test.test_tkinter.test_widgets.EntryTest)
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Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/test_tkinter/test_widgets.py", line 333, in test_insertborderwidth
 self.checkPixelsParam(widget, 'insertborderwidth', 0, 1.3, -2)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 169, in checkPixelsParam
 conv=conv1, **kwargs)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 59, in checkParam
 self.assertEqual2(widget[name], expected, eq=eq)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 43, in assertEqual2
 self.assertEqual(actual, expected, msg)
AssertionError: 0 != 1
======================================================================
FAIL: test_bitmap (tkinter.test.test_tkinter.test_widgets.LabelTest)
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Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 240, in test_bitmap
 errmsg='bitmap "spam" not defined')
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 75, in checkInvalidParam
 widget[name] = value
AssertionError: TclError not raised
======================================================================
FAIL: test_bitmap (tkinter.test.test_tkinter.test_widgets.MenubuttonTest)
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Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 240, in test_bitmap
 errmsg='bitmap "spam" not defined')
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 75, in checkInvalidParam
 widget[name] = value
AssertionError: TclError not raised
======================================================================
FAIL: test_bitmap (tkinter.test.test_tkinter.test_widgets.OptionMenuTest)
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Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 240, in test_bitmap
 errmsg='bitmap "spam" not defined')
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 75, in checkInvalidParam
 widget[name] = value
AssertionError: TclError not raised
======================================================================
FAIL: test_bitmap (tkinter.test.test_tkinter.test_widgets.RadiobuttonTest)
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Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 240, in test_bitmap
 errmsg='bitmap "spam" not defined')
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 75, in checkInvalidParam
 widget[name] = value
AssertionError: TclError not raised
======================================================================
FAIL: test_insertborderwidth (tkinter.test.test_tkinter.test_widgets.SpinboxTest)
----------------------------------------------------------------------
Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/test_tkinter/test_widgets.py", line 333, in test_insertborderwidth
 self.checkPixelsParam(widget, 'insertborderwidth', 0, 1.3, -2)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 169, in checkPixelsParam
 conv=conv1, **kwargs)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 59, in checkParam
 self.assertEqual2(widget[name], expected, eq=eq)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 43, in assertEqual2
 self.assertEqual(actual, expected, msg)
AssertionError: 0 != 1
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The following test failures are observed with Tk 8.4 on OS X (python.org 32-only installer with ActiveTcl 8.4.20 on OS X 10.5.8):
======================================================================
FAIL: test_debug (tkinter.test.test_tkinter.test_text.TextTest)
----------------------------------------------------------------------
Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/test_tkinter/test_text.py", line 22, in test_debug
 self.assertEqual(text.debug(), 0)
AssertionError: '0' != 0
======================================================================
FAIL: test_insertborderwidth (tkinter.test.test_tkinter.test_widgets.EntryTest)
----------------------------------------------------------------------
Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/test_tkinter/test_widgets.py", line 333, in test_insertborderwidth
 self.checkPixelsParam(widget, 'insertborderwidth', 0, 1.3, -2)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 169, in checkPixelsParam
 conv=conv1, **kwargs)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 59, in checkParam
 self.assertEqual2(widget[name], expected, eq=eq)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 43, in assertEqual2
 self.assertEqual(actual, expected, msg)
AssertionError: 0 != 1
======================================================================
FAIL: test_insertborderwidth (tkinter.test.test_tkinter.test_widgets.SpinboxTest)
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Traceback (most recent call last):
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/test_tkinter/test_widgets.py", line 333, in test_insertborderwidth
 self.checkPixelsParam(widget, 'insertborderwidth', 0, 1.3, -2)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 169, in checkPixelsParam
 conv=conv1, **kwargs)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 59, in checkParam
 self.assertEqual2(widget[name], expected, eq=eq)
 File "/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/tkinter/test/widget_tests.py", line 43, in assertEqual2
 self.assertEqual(actual, expected, msg)
AssertionError: 0 != 1
----------------------------------------------------------------------

There are three different classes of errors here:
1. test_debug AssertionError: '0' != 0
This has already been fixed by 7dab4feec126/05e84d3ecd1e/e7d922d8ee03 for Issue6517.
2. test_bitmap checks for invalid bitmap values do not fail as expected with Cocoa Tk
This appears to be a bug in Cocoa Tk, reproducible with the wish shell. I've opened a Tk issue for it:
https://core.tcl.tk/tk/tktview?name=31cd33dbf0
The simplest approach would be to just skip test_bitmap for OS X until this is fixed. A more focused workaround patch would be to only skip the checkInvalidParam test when running under Cocoa Tk:
 'aqua' in root.tk.call('tk', 'windowingsystem') and
 'AppKit' in root.tk.call('winfo', 'server', '.')
3. test_insertborderwidth failures for Entry and Spinbox widgets
After a little investigation, I think this is a test case error. at least for Entry. According to third-party Tkinter documentation (http://infohost.nmt.edu/tcc/help/pubs/tkinter/web/entry.html) for the Entry widget:
"By default, the insertion cursor is a simple rectangle. You can get the cursor with the tk.RAISED relief effect (see Section 5.6, 'Relief styles') by setting insertborderwidth to the dimension of the 3-d border. If you do, make sure that the insertwidth option is at least twice that value."
It looks like for both the OS X Carbon (8.4) and Cocoa (8.5 and 8.6) Tks, the restriction about insertwidth is enforced, but with the X11 Tk, it is not.
Some examples using Cocoa Tk (8.5.15) and 3.4.0b3:
>>> from tkinter import *; r = Tk()
>>> w = Entry(r, insertborderwidth=1.3); w.pack(); w['insertborderwidth']
0
>>> w = Entry(r, insertborderwidth=1.3, insertwidth=10); w.pack(); w['insertborderwidth']
1
>>> w = Entry(r, insertborderwidth=2); w.pack(); w['insertborderwidth']
0
>>> w = Entry(r, insertborderwidth=2, insertwidth=10); w.pack(); w['insertborderwidth']
2
>>> w = Entry(r, insertborderwidth='10p'); w.pack(); w['insertborderwidth']
0
>>> w = Entry(r, insertborderwidth='10p', insertwidth=10); w.pack(); w['insertborderwidth']
5
Using X11 Tk (8.6.1) on OS X:
>>> from tkinter import *; r = Tk()
>>> w = Entry(r, insertborderwidth=1.3); w.pack(); w['insertborderwidth']
1
>>> w = Entry(r, insertborderwidth=1.3, insertwidth=10); w.pack(); w['insertborderwidth']
1
>>> w = Entry(r, insertborderwidth=2); w.pack(); w['insertborderwidth']
1
>>> w = Entry(r, insertborderwidth=2, insertwidth=10); w.pack(); w['insertborderwidth']
2
>>> w = Entry(r, insertborderwidth='10p'); w.pack(); w['insertborderwidth']
1
>>> w = Entry(r, insertborderwidth='10p', insertwidth=10); w.pack(); w['insertborderwidth']
5
So it appears the tests could be made to pass on OS X Cocoa and Carbon Tks and on X11 Tks if insertwidth is also set on the insertborderwidth tests. That would need to be tested with Windows Tk.

Thank you Ned for your great investigation.
Here is a patch for these two bugs.
A test for invalid bitmap is skipped (also added a test for bitmap specified in external file). For insertborderwidth tests the insertwidth value is explicitly increased (yes, the Tk source code bounds insertborderwidth by insertwidth/2).

The supplied patch fixed all of the failing test cases on OS X, tested on 3.3 and default with Cocoa Tk 8.5 & 8.6, Carbon Tk 8.4, and X11 Tk 8.6. I'll test on 2.7 after the modified patch is applied. Thanks, Serhiy!

New changeset 129eb818d9b2 by Serhiy Storchaka in branch '2.7':
Issue #19761: Fixed Tkinter tests on OS X.
http://hg.python.org/cpython/rev/129eb818d9b2
New changeset 76cd2e59b13d by Serhiy Storchaka in branch '3.3':
Issue #19761: Fixed Tkinter tests on OS X.
http://hg.python.org/cpython/rev/76cd2e59b13d
New changeset 2efe0778a4cf by Serhiy Storchaka in branch 'default':
Issue #19761: Fixed Tkinter tests on OS X.
http://hg.python.org/cpython/rev/2efe0778a4cf 

Thank you Ned.