What kind of router are you using? (Identifying this will give us the default IP address of the router, as different manufacturers use different addresses)...
IMPORTANT
What I would do, is connect the GOOD computer to the router, and get to the router's CONFIG page... Check to make sure that MAC ADDRESS FILTERING is NOT on, and if it is, make SURE that the 'bad' PC's MAC address is in the list. My bet based upon the symptoms is that MAC address filtering is enabled, and that is why you can't connect... (I actually had this happen to me a week ago when I got my fiancee a new laptop and couldn't for the life of me connect it to the network... lol.)
If that didn't solve the problem.... the do this... when the 'defective' computer is connected to the router (hopefully with a KNOWN GOOD cable) what output do you get when you perform an IPCONFIG command in a command window? (START > RUN > type 'CMD' > type 'IPCONFIG'
It is either going to be a 192.168.x.x or 169.254.x.x address depending on if the router has a default IP scheme, and if the computer can see the router or not.
Have you tried to un-install and re-install the NIC on the 'bad' PC?