The smallest beam size made can fail under it's own weight.Yes it is possible. See dayhiker's post above. I don't remember the size but it was pretty darn big.
If I check beams under their own weight and consider a stress ratio of 2.0 to be about where it might fail, and calculate the length needed to get to that stress level, then:
W6x9 - 23' - the beam would deflect 1.19" right before failure.....assuming that's about the stress level where it would collapse.
W14x22 - 60' - the beam would deflect 1.11"
W44x335 - 155' - deflection = 4.8"