Neal Stephenson wrote a bit about this in Cryptonomicon; laying new undersea cable is both expensive and time consuming, but the cost of cutting existing ones is fairly low. So if any sufficiently-funded individual, corporation, or nation-state wanted to hold a gun to the world's head, cutting undersea data cables wouldn't be a bad way to do it.

The problem is, you can't make that kind of threat in a subtle way, so to consider something like it you would have to be some kind of international pariah with a warmongering streak and a history of 'lying in plain sight' about your own nefarious deeds.

Edit: Okay, we're in better shape today than we were in the '90s and cutting off Cyprus' internet wouldn't cripple the world, but we still don't have THAT many cables running across the Atlantic and Pacific.