Checked with our admin guys, and I ended up posting a question on Experts Exchange to see what they could come up with.
Well, in the end, someone there pointed me to the faulty Intel drivers link above and sure enough, with "ipconfig /all" I got all zeros for my MAC address. Apparently you can get an IP in DHCP with an all zeros MAC - but all the PCs with this MAC address get the same IP.
OK, the link says to load the old drivers. I browse to my motherboards website, sort to the NIC drivers, then XP and scroll down and...ummm....ok. The Intel guy on the Intel site said to download and install the older driver. Wouldn't you think they would have an older driver on their site? Nope.
So, a word of caution before you update your XP Intel NIC drivers - don't!