If it occurs on both Wi-fi and CAT5 your almost certainly looking at a settings issue in Windows or your router vs. hardware.
Could be quite a few things, though.
I would assume this is a computer running XP.
Worse comes to worse you could always re-install XP to reset any bad settings, but I'd try to resolve them first. Also could be settings in the router. Maybe MAC Address filtering.
If you can find your Network Connection settings (don't remember where they would be in XP), right click the connection you are using and then click Properties. Select the IPv4 settings, and click Properties. Make sure that is set to Obtain your IP address and DNS settings automatically.
To make sure it's not a DNS issue you can just type an IP address into the browser. Try Googles IP:
http://74.125.224.72/IF that works, you've got a DNS issue somewhere.
Another troubleshooting option (as well as a temporary fix) would be to boot the computer to a LIVE CD of Linux. Ubuntu is pretty easy. Download and burn a CDR of the install disc and have the computer boot from the CD. You won't be able to save any settings from there, but if you can get on the internet that way you've eliminated the router as the problem. For Ubuntu your going to want the 32 bit version from here:
http://www.ubuntu.com/download/desktopThat's all I can think of for now.