I think the most unrealistic bully characters are the ones that seem strangely isolated. Like everyone lives in fear of them, no one laughs along with them when they're picking on someone. I guess it could happen, but in my experience that's rare.
The typical bully thrives on social approval (some even bully specifically because they crave that attention from others, sort of like a class clown). They put on a show for everyone else; they say and do the mean things that other people would say and do to their target if they didn't have kindness and/or restraint.
And even if they do the actual work of bullying alone, they usually have a small group of friends (their primary 'audience') that they at least talk to about their targets,
The power hierarchy, in brief:
Level 1: People the bully really likes/respects. They actually control the bully's actions indirectly; if something the bully does doesn't get a satisfactory reaction from them, they may stop doing it. (optional)
Level 2: The bully, along with fellow bullies that collaborate with them of their own volition.
Level 3: Popular non-targets. They may or may not approve of the bully's actions, but in any case they are considered too well liked to be targets most of the time. When they do clash with the bully, it's usually very dramatic.
Level 4: The bully's 'pets', minor bullies that follow the main bully's lead and are essentially controlled by them. If they step out of line, there are usually immediate consequences from their peers or from the main bully themselves...and in a large group, at least one of them usually becomes the main bully's 'practice dummy', someone they ("jokingly") abuse when their usual targets are not around. (optional)
Level 5: Regular non-targets. The bulk of the 'audience', made up of people who either escape the bullies' notice most of the time or fly under the radar completely. They may or may not care about the bully's behavior...in any case, they usually avoid the targets, or even shun them, just to stay safe.
Level 6: The targets. Anyone who is low enough in the social hierarchy that you can abuse them and get away with it (i.e. the majority of those in power will not care). If you wanna get creative with this, see how high up you can push the target on the social ladder, yet still have them be bullied.*
In conclusion, as long as you have all the essential tiers of this hierarchy working and interacting throughout the story, whatever you have going on will probably look realistic.
And they don't all have to be named characters or anything. It should just be clear that they exist, through the descriptions of the bully and target's lives.
*Be careful with this-- don't fall into the typical 'Mary Sue' trap of having a character who is bullied for basically no reason other than sympathy points.
Yes, anyone can be bullied, but it has to fit the environment...or more specifically, the bully. If they are an adult 'womanizer', they will probably not bully a girl who is at least pretty and has lots of friends...they would prefer to 'perform' for them, and said girl would end up on Level 1 or 3.
To give another example, if they're a typical 'mean girl' bully, they would definitely bully said girl (they would see them as competition)...UNLESS, said girl made attempts to get on her good side. If she's pretty and popular enough (and if her morals are loose enough...), she could easily get absorbed into the bully's clique, and/or find herself on any level from 1 to 4.