The problem was sporadic.
Finally, I started to figure it out. The Last.FM client, which is always running in the background when I work, was the culprit. It was causing the terminal to lose its focus.
Ironically, I discovered the problem when the client was doing a self-update and shut-down. Then, X11 started working again. The new client (1.4) seems to not be conflicting with X11. But I'm not counting on that.
My suspicion is that other programs are having similar effects.