I was horrifically bullied at school, for the entire eleven years. When I went to a school reunion one of my worst bullies came up to me and said how great it was to see me, was incredibly nice and friendly, and insisted on buying me a drink. I think it was her way of saying sorry. Another woman at the same reunion said that someone else had wanted to attend but couldn't, and she was really desperately sorry to miss the union because she'd hoped I'd be there. She wanted a message passed on to me, that she had never forgotten how horrible she'd been to "that poor, poor girl" (ie, me) and that she was truly sorry - so yes, in my experience, some bullies do feel remorse when they grow up.
On the other hand, another group at that same reunion recognised me, and said things along the lines of "Oh, look who it is! We remember you! Everyone always laughed at you! You didn't have any friends, did you?" I moved away from that group at top speed - it was most odd and disconcerting, as if they hadn't moved on at all in twenty-five years, and were right back in the playground.
(If you're all wondering why I went to a school reunion at all, it was because my best friend from school, who is still a friend, talked me into it, and was there for moral support. I'm actually glad I went, as it was interesting.)