I am here to say that I too cannot play piano and talk at the same time. I cannot sing along with my playing either. but as LAB pointed out, its only when im playing both chords and lead. if I do talk or sing, I end up playing off beat and hitting the wrong keys. which is why I usually ignore people while playing.
with guitar, where its only chords, I can sing along and can probably even give a shiur while playing.
someone should do a research paper on this fascinating phenomenon.