Say a student was insulting another or some individual. If the remarks aren't categorized as a death threat, would the "bully" be suspended? Also, what if this was a case of Cyber Bullying, where the problem didn't occur on school grounds? Does a school administration have the jurisdiction to punish students who make insulting remarks on the internet?
The school must comply with relevant legislation and regulations within its jurisdiction.
Unless there is something specifically forbidding their suspending a student who makes insulting remarks on the internet, they will generally be permitted to take reasonable disciplinary action.
In the United States, education is largely handled by the States. A similar, but broader question, concerning suspensions is this one by jimsug. In it, you'll find a helpful compendium of U.S. student disciplinary laws.
As an example, Illinois recently allowed suspension for bad behavior via electronic means.