The ability of a machine to ping itself is just showing that the loopback IP device , 127.0.0.1, is working. It means that the packet goes through the protocol to the client and back - it does not mean that the packet is necessarily going out to a network.
If you please I would recommend starting by rebuilding the network stack. There is only one way to force Windows to rebuild it completely without errors - and it is not the commonly known way. Have the drivers for the NIC handy just in case:
Start the computer in Safe Mode, go into Control Panel / Network and remove everything. Do not reboot when asked.
Go into Control Panel / Add-Remove Programs / Windows Setup / Communications and uncheck Dial Up Networking, Dial Up Server and Vitural Private Networking if you have them installed. Now reboot.
The computer will rediscover all hardware. Do not reboot when asked. Go into Control Panel / Network and Control Panel / Add-Remove Programs / Communications (both areas if need be), install and properly configure all network components that you need. Now reboot.
Does WINIPCFG show that the set IP's are on the same segment? (That is, are the first 3 tuples the same? If one is 192.163.0.1 then set the other to 192.163.0.whatever). Is the IP mask the same?
Are you trying to ping using host names or IP addresses? If using hostsnames then the HOSTS or LMHOSTS files may have incorrect entries. Check. Your mileage may vary...