Okay Guys, here's where we are.
The Athalon can ping the Xeon by name or ip address.
The Xeon can ping itself by name or ip address, but NOT the Athalon.
Neither of these computers is on the web or will ever be.
Both computers are connected by a brand new Lynksys Hub with brand new ethernet cables.
Both external and Windows Firewalls are TURNED OFF on both Computers.
Under "My Computer" properties, Computer Name shows the name of each computer, the correct workgroup name, and the fact that they belong to a small network with NO DOMAIN.
The Athalon shows all the shared files, including the drives of itself and the Xeon in the Network Places page.
The Xeon shows the icons for both PC's in the workgroup. But I can only enter the Xeon's icon, not the Athalon. Conversely, on the Network Places page of the Athalon I can only access the files in the Athalon. I get an error message when I try to open one of the Xeon files from the Athalon.
The problem has now been determined to be with the Xeon, since the Athalon has no problem pinging the Xeon. However when asked to view all the pc's in the workgroup, the Athalon only sees itself.
I uninstalled the Ethernet Card virtually to see if a sort of rebooting of the card would clear the blockage. No change.
I don't feel so inept anymore, since every IT person I've spoken to so far has commented on how unusual this scenario is, and nobody yet has cracked this mystery.
Any other suggestions?