Considering most of my uniwork goes through the internet, a week before I started to fall behind seriously.
Excluding school, a month. I've gone that long before. I have quite a few singleplayer games to entertain myself with, and the 5 hours per day I spend online would do wonders for my guitarplaying. That said, I get all my news and such through the internet, so it'd be like living under a rock. Newspapers are great and all that, but quite expensive when you're already barely surviving with a student budget. The internet is great in that way; easy access to the rest of the world.
With enough money to buy new singleplayer games (assuming everything doesn't go over to Ubisofts Nazi-DRM), and a way to get news from the world, I could probably go without it for years. It'd take some time to adjust, but entirely doable. Would probably cave in when ME3 is released and I needed the DLC for it.