Microsoft keeps finding new and improved ways in which to be annoying.
Windows XP Service Pack 3, it turns out, breaks Remote Desktop access for some strange reason. Once you know the solution, it’s not a terribly difficult fix (see below) — but you gotta wonder how it ended up being broken in the first place.
Cutting to the chase, here’s the fix. Add the following registry key:
[HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Session Manager\Memory Management]
(Disclaimer: If you modify your registry, you imply that you know what you’re doing and want to do this. This mod worked for me; YMMV.)