Unlike GzipFile the BZ2File does not have defined name attribute. The reading of this attribute results in: "AttributeError: 'BZ2File' object has no attribute 'name'".

This is truly missing feature and wrong behaviour, compare:
https://hg.python.org/releasing/3.4.3/file/933c3041d548/Lib/gzip.py
and 
https://hg.python.org/releasing/3.4.3/file/933c3041d548/Lib/bz2.py

In addition to that the BZ2File did have name attribute in python 2.7.* .

In attachment there is tar ball with example files, expected results in python 3 and results in python 2.

I'd be happy to take a look at this one, if no one else is working on it.

I've got a fix for this now.  In working on it, I've discovered that not all file-like objects have a .name attribute.  io.BytesIO (which is used all over the test suite) does not.  I've written a patch that always creates a .name attribute on BZ2File, but sets it to None if the associated file-like object has no .name.

See the linked branch

It's probably too late for this to get into 3.5, since we're already in betas.  Pushing back to 3.6.

Patch adds a bz2.BZ2File.name property corresponding to the filename passed in.

Please may you add a test that uses "io.BytesIO()" as the filename argument.  BytesIO() objects do not have a 'name' attribute

The https://bitbucket.org/cliff/cpython#python24258 URL 404's

Looking at the attached bz2.py diff, I would change:

         if isinstance(filename, (str, bytes, os.PathLike)):
             self._fp = _builtin_open(filename, mode)
+            self.filename = filename
             self._closefp = True
             self._mode = mode_code

to special-case os.PathLike and do:

            self.filename = str(filename)

I would expect BZ2File.name to be a string.   Returning a PathLike means lots of legitimate string methods will not work on it.  Also, it should be "name" as an attribute, not "name()" as a method, to match other existing interfaces.