Oops, forgot the interesting part!
When I logged back into my RDP session, all of the keyboard smashing I had done was there!
The graphics lock up but the machine is still functioning in some ways. It was sending my keystrokes to the remote server even though the graphics had stopped displaying anything. Also, this was the only time we've seen or heard of the graphics going out in stages. The keyboard input stopped displaying onscreen for several seconds but the mouse movement still displayed for a few seconds after that before it, too, froze entirely.