I say 1980s, and here's why:
1- THE CARS. The newest cars look 1980s, and the oldest look 50s and 60s - and there's a lot of 50s and 60s cars around. If the game took place in 2006-2007, that would mean the newest cars in the game are 20-25 years old, and the oldest cars are about 50 years old. There isn't a town in America where there are absolutely no cars newer than 20 years old. The newer cars (mid 70s and newer) are often seen in Bullworth Vale and the town center, whereas the older metal (pre mid 70s) are often seen in New Coventry and the industrial area. A date of, say, 1984-1985 would mean that the newer cars in the game are really newer cars, and the older cars are just old cars, not the classics they are today.
2- The cliques. The "greaser" ideal had its peak in the 80s, as did the "preppy" clique ideal. Nerds and jocks are also archetypal 80s cliques.
3- Technology. Arcade games are common. In 2006, arcade games weren't nearly as ubiquitous as they were in the 1980s. And while the games have graphics that are too good to be 80s, the concepts were definitely around back then. In addition, there's not a cell phone in sight in the game, the computers in the library are VERY old (early 80s or even 70s!) and about the only new technology in the entire town is a laptop in a basement.
4- Other elements around the town. For instance Happy Volts Asylum would never be open as late as 2006. That's a mid-20th century asylum that would have been badly outmoded by the 1980s. And badly outmoded it was. I would imagine HVA wouldn't have lasted past the end of the 1980s.
I envision Bully to have taken place in 1984-1985 school year. I have a whole fan storyline I wrote long ago about the history of Bullworth and Bullworth Academy.