Two Plus Two Older Archives - internet problems due to old computer?

I'm having trouble with my cable modem and it's costing me money because i'm always disconnecting during poker games. The computer I use is a piece of crap from 1998 that hardly works any more. I think it's a pentium II with a 6 GB hard drive. Is it possible for this old computer to cause problems with the connection simply because it's old? It disrupts the connection on the other computer in the house which is a newer computer. Would it help if I switched the modem and the router to the new computer?

As you can tell, I really don't know much about this stuff. But if it's possible that the old computer is causing the disruptions, that would be a great excuse to get a new computer.