Not anymore... there's only one Internet network (except for Internet2, but that doesn't count yet). But your friend is right too.
The nifty thing about routers is that they can easily be trained and/or forced to route things the way you tell them to. If certain hubs (cities) become unusable, you just adjust your routing tables to not use those places. Most of this happens automatically nowadays anyway. The protocol is perfectly designed to be able to heal communications around a sudden "hole" in the web. But even a web can be brought down if you hit all of the nodes at the same moment.
So much of the web anymore runs through and depends on the backbones of only a few major providers. In fact there was really ever only one true Internet backbone, and that was UUNet. Then MCI bought them out. There's a lot of fiber running all over the country, but for the most part it all runs together in the same trenches, and most of that runs the major highways. For example... take out I-90 at Chicago and (I-80?) in Denver, and suddenly the East coast can't talk to the West Coast.
I remember part of his discussion was about rerouting the hubs. On the fiber issue, couldn't that be rerouted through regular phone lines and satellites? I'm sure that'd slow things down, but I'm thinking in terms of the military still being able to communicate if necessary. I'm also wondering if there's buried cables branching off from the main ones and bypassing the highway chokepoints in case of that contingency.