Update
I have new symptoms.
The problem is "spreading".
Recently, a crazy lightning storm that came through a modem knocked out several pieces of equipment (it actually knocked out a bunch of equipment around town, according to my contacts). It seems the strike propagated through the modem, out its ethernet ports, and then knocked out several network devices. It took out an ethernet port on my gateway, completely killed a VoIP box, went back out through that box's ethernet, knocked out several switches, and from there also damaged some of the ethernet ports on my workstations, as well as some printers.
Now, I shouldn't get too off track with the storm, because it doesn't have anything to do with the problem that this question is about, which predates the storm, but the necessary repairs did reveal new symptoms.
After replacing all the damaged switches, I had everything mostly back up and running. But the workstations with new network cards are not getting DHCP from the Windows Server.
On one computer, I am able to do a unique test, because the onboard NIC still partially working but is unstable (because of the lightning strike). It gets DHCP but then loses the connection for a bit and then reconnects. I installed a new NIC and the connection is rock solid (I tested this by connecting it directly to the modem: see point 6. above), but when connected to the internal network, I get no DHCP from the server as in the original symptoms described above.
So, the computer gets DHCP when using the original (now damaged and flaky) NIC, but not when using the new NIC.
Is there something with Windows Server that would make it refuse to hand out an IP if it doesn't like the NIC for some reason?