Consider the attempt last year for the government of Egypt to block cross-border Internet traffic.   Sites like gmail, Youtube and CNN even were not accessible by those within Egypt.  But the traffic within the country was still functional.  Eventually the bitcoin client would find peers within the country (or helped a bit by manually specifying which peers to connect to) so all that needs to happen is for one of those nodes to connect to the rest of the internet for the blockchain to be kept current.  That could occur over dialup if nothing else.

There are other ways the Internet could survive if disruptions to massive subcomponents were to occur (e.g., if consumer access providers were to block all traffic, for instance).  Metro-wide networks built using mesh networking technologies with inter-city peering links could some day make reliance on the existing Internet infrastructure.  You wouldn't get Facebook but you could still route e-mail, transfer data files and transact using bitcoin.

Here's one approach for a quick-deploy emergency internet:
 - http://wiki.hacdc.org/index.php/Byzantium#Project_Goals