The same thing is going on with my son. I know that a lot of times ADD gets confused with giftedness. I tried to work with the school to move him up and see if having something more interesting to learn would help with the behavior but the school said until his behavior improved they weren't going to do anything.
We noticed at home that he was having trouble focusing long enough to complete a sentence and that he couldn't do almost anything we asked him no matter how hard he tried because he couldn't focus long enough.
I took him to get evaluated and the psych said that he had ADD. We started medicine and he's been a different child at school. I'm not saying that being bored didn't have anything to do with it though. In fact the psych said that if the school ever does give him work on his level we should try to take him off of the meds and see how it goes.
It's all very confusing to me because the symptoms of ADD are so close to the traits of a bored gifted kid but for now the medicine was our only option and it seems to be going well.
Good luck!