Thanks, Trouble.
All machines are running Windows 7 Ultimate. All machines can ping each other, no problem.
Per your suggestion, I've checked the inbound rules for Remote Desktop Connection. I see one private profile and one public profile. They are both enabled and set to "allow the connection." I also disabled the firewall, the antivirus, etc.
Still the same problem.
Here's another clue that might help narrow things down. I also tested it with other Windows 7 Ultimate client machines on the network. All the other machines can RDC to each other, no problem. The DESKTOP machine can RDC into all the other machines, no problem. But none of the others can RDC to the DESKTOP machine.
What do you think? What should I try next?