I think it can. Some atonal pieces can sound very complex and disordered with regards to tonality to a person that hasn't been exposed to it. However, once a person becomes familiar with it, it is possible to find some order in it and a potential tonal center.
However, before going further, one must understand that the terms "tonal" and "atonal" are extreme opposites. Many compositions that people may call "atonal" are really in the middle somewhere, which might mean that the tonal center might be harder to spot, but is still there. Even for a really complex piece like a composition by Finnissy, one might be able to argue that it has a tonal center somewhere (I'm not saying anyone would take your word for it).
That's enough ranting for me. Point being, I think one can argue a piece's potential tonality if they are mor familiar with the genre.