nah. bullies are people that abuse some form of power.
i mean, they must have some form of power/influence, right? otherwise, how could they bully anyone?
either they're the strongest kid in their class or their parents are the richest ...or they have actual anger issues (but those people need a psychologist, not my pity).
i could counter your anecdote with one of my own, where a guy with abusive parents is more helpfull and kind than others because he knows how shit it feels, but anecdotes aren't worth much ;)