Yes, renters insurance covers burst pipes. “Accidental discharge or overflow” of water from your plumbing system is specifically listed as a covered peril in standard renters insurance policies, and burst pipes count as a type of “accidental discharge.”
Renters insurance also covers frozen pipes, so if your pipe freezes and subsequently cracks or bursts, that’s covered, too.
Pipe disasters that aren’t covered
Although burst pipes are almost always covered by renters insurance, sewage and drain backups usually aren’t. If you’re worried about those disasters, consider purchasing water backup insurance for renters, which most companies offer with an affordable rider (an addition to your existing renters insurance plan).