Not necessarily. I work with a 7 year old kid who is emotionally disturbed. The other kids will still play with him when he's not angry or screaming and yelling, even after they've seen him hit and bite their friends.
They don't bully him, he bullies them. He needs help and the school district is passing the buck to save a buck.
Oh, and this 7 year old has already said he'd like to set the school on fire so he can burn up another student I work with (who is a sweetie).
I can see where this could occur.
It was also my experience that those who bullied in school were also secretly or openly bullied by others, and so they did the same, basically just passing it on down the line.