I find it's usually the bullies who are the most insecure.
I would miss months of school and then return with bright blond hair. Needless to say, there was bullying. I wasn't beaten up daily, but there was name-calling and jealousy. You have to bear in mind that 'Harry Potter' wasn't cool. I wasn't part of the 'Terminator' franchise.
Draco's not really a bully. He's not exactly the biggest, strongest guy in the world. He's more a rich, snobby person. He thinks of himself as really cool.