Just a quick question; you’re not using any networking software provided by the ISP, are you? - your connection between computers and router is mediated entirely by standard networking protocols, isn’t it?
I do have a seperate router, a Belkin, as mentioned in the OP 
Mangetout, I know, that’s why all the question marks after the power comment. Everything I have has it’s own power source and the data merely travels between them. The modem and the router are plugged into their own power supplies, of course, and the network card gets it’s power from the PC.
I can’t imagine what sort of power level the tech was referring to, I suppose I should have opened that can of worms and asked, but I wanted to wait till I got home and was sitting in front of the computer before I asked too many Q’s.
I still haven’t had the time to do so. Probably not till the weekend now. Time is tight at home with fall approaching.
And no, I’m not using any sort of RR software. Everything was set up manually.
I thought about installing the RR connection software that came with the modem but was talked out of it by a PC tech here at work. He, by the way, is stumped also.
Jayrot, you’re right, it is request denied. But remember, I have full access to any place I want to go for about 5 minutes, then I’m cut off from the web. I can’t even go to Google.com and I can’t even refresh the same page I was just on, or hit the back button without getting the message.
I can only imagine he was either just talking rubbish, or was thinking out loud about the network to which your modem is attached.
Cable modems are a lot like the old 10Base networks that ran across coaxial cabling; power levels and upstream/downstream signal-to-noise ratios are important and when they go wrong, you can end up with a flaky connection. The cable modem is the device that handles all of that stuff, and turns it into an ethernet connection that you plug your router into.
There’s just no sane way that connecting two computers to the router would do anything to the power levels on the cabling that is the other side of the modem. What he said just doesn’t make any sense at all - plugging more cable modems into the same segment will affect the power levels, but that’s nothing like what you’ve done.
Try starting your computer and letting it sit for 10 minutes then access the internet.
Do you get access that dies after 5 minutes?
That’s something else I need to try.
Also another thing to test would be whether after your internet connection has “died”, you can still access files on the other computer (through Windows file sharing). This would determine if the issue is one with the internet (and thus dsl/cable modem) or your local network infrastructure.
Oh, and another thing to keep in mind is that Windows 98 sucks. 
I had Roadrunner issues once with 2 computers not getting online, I think it was mostly an ethernet cable problem BUT the guy did come to my house and test the signal and say it was below the recommended power, which is why the connection was iffy. So to fix it he changed the splitter on the incoming cable so that the signal going from the wall straight through the splitter went to the modem, and the one perpendicular was the TV.