Internet Explorer Prob

I have a computer that is part of a network, and in having problems on the internet. The network is on DSL. All of the computers have there own IP. When I boot the computer up, it works online for a few minutes, then will not find any website. However, I can still ping the address of all of the other systems on the network and the address of the router successfully. I’m not getting it. It happens with 2 or 3 systems on the network and no others. some of the pc’s are running 98SE some are on XP. If you reboot the system then it works again for an unknown amount of time; it changes from time to time. Help 🙁