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Right, the HTML example was always a bit cutesy (hence the disclaimer in the header), but it's hard to come up with a good illustrative example that isn't already a native context manager in the standard library.
Perhaps it would make sense to change the example text completely, and write something entirely abstract like:
 from contextlib import contextmanager
 @contextmanager
 def managed_resource(*args, **kwds):
 # Code to acquire resource, e.g.:
 resource = acquire_resource(*args, **kwds)
 try:
 yield resource
 finally:
 # Code to release resource, e.g.:
 release_resource(resource)
 >>> with managed_resource(*args, **kwds) as resource:
 ... # Resource is released at the end of this block,
 ... # even if code in the block raises an exception
Then the introductory text could be updated to say something like "While many objects natively support use in with statements, sometimes a resource needs to be managed that isn't a context manager in its own right, and doesn't implement a ``close()`` method for use with ``contextlib.closing``.".