This sounds like the Y2K problem, only a lot more realistic.  According to PC Magazine’s Lance Ulanoff, we may be just a couple of years away from the end of the Internet — or at least the end of its expansion.  It’s not from lack of interest, but a built-in limitation of the addressing system known as Internet Protocol, or IP:

Like Y2K, the solution is easy, but getting people to spend money on it is the problem. It took a lot of hyperbole and sky-is-falling hysteria in 1999 for users to finally get rid of their old DOS 3.0 machines and old database systems that couldn’t recognize a year that didn’t start with 19. Unfortunately, the pressure this time won’t be on the people with IP addresses already, but the people who want new addressing.

Maybe the Internet will be a club — closed to new membership, only allowing replacements for those who opt off the system or die. It’ll be a case of who you know. That means … you’d better start being really, really nice to those of us with our own IPs.

Update: As always, South Park has foreseen the catastrophe (h/t RightWinged):