Basically, there technically IS internet. It's not the internet we think of today with websites but rather instant communication relays with the ability to send photos and text.
A Network, an internet (notice small i), and the Internet (Capital I) are different things. Fallout has shown for the most part networks, and perhaps on enterpretation internets (internetworked networks), but no "Internet".
Your Works LAN - Network.
Using your ATM card in another country - uses an internet, but not the "Internet" (the host bank connects to your bank via an "interbank network" {eg CIrrus for Mastercards and Maestro cards, Visas use PLUS}. Networks connecting to networks is an internet, but this is seperate to the "Internet")