I've seen just as much bullying at elective academic summer camps (e.g. Space Camp, JHU CTY), as at public schools. Both as a student and as a teacher. I don't agree that it's necessarily a product of the American public school system.

Particularly, in my experience, bullying tends to be exacerbated during periods of social uncertainty, such as a change in the class cohort. The bullies feel vulnerable and are looking to prove themselves, and the bullied are vulnerable, having not yet had any chance to form social support groups.