If what Kaolian suggested doesn't work, it's posible that your ISP authenticates by MAC address. I know it was pretty common practice down here for a while, but I haven't done domestic IT work in a while, so it's been a while since I've had to mess with this kind of stuff.
If they do, than that one computer will be the only one that works with your internet. However, your router should have a feature built into it to trick your modem. Linksys calls it MAC cloning. I'm not sure what brandname router you have, but it should be something like that.
On the computer that always works, go to start, pick run, type cmd and press enter. That should bring up a black dialouge box. In that box type "ipconfig /all" It should bring up something that looks kinda like this:
http://omegabrian.googlepages.com/macaddy.JPG
that line that I marked will have your MAC address. It will be 6 pairs of numbers and letters. write it down.
Next, log into your router and find where you can assign it a MAC address. That should be specified in the manual for your router. I took a page out of the manual for the Linksys WRT54GL for an example of what you should be looking for:
http://omegabrian.googlepages.com/macclone.JPG
IF your ISP is authenticating via MAC address, this should effectively trick your modem into thinking the original computer is still hooked straight up to it. I'd supply more detailed instructions but it will vary widely depending on the make/model of your router.
Good luck :D