As it exists today, "the Internet" seems like a kind of magic (e.g., people now talk about storage of data "in the Cloud"), but the Internet really is and remains just a network of physical boxes (servers) and physical wires.
Say a number of Luddite commando teams are raised. Armed with plain old-fashioned hammers, they are dispatched to smash up the world's limited number of servers and dynamite major fiber-optic cable chokepoints. Couldn't doing so "destroy" the Internet?
The answer: The physical Internet in the mid-2010s may still be smaller than a single oil tanker (maybe two, by now. The original answer was written in 2012). An oil tanker is a just few football fields in length, and a whole lot of the world's servers are together at few central locations, or so is my understanding. This wouldn't require so much smashing, after all.