I had gone home somewhat early yesterday to call tech support before taking the computer in for service. But when I checked to make sure that the computer was still showing problems, it booted up successfully on the second attempt and kept working for several hours.
It locked up at the end of the evening, though, and wasn't doing well this morning, so I called tech support.
I carefully described the symptoms we had been having, and how when it froze up there was no error message, and control-alt-delete didn't work. The support guy listened to my problems with the occasional clarifying question, instead of frog-marching me through a scripted dialog.
After my description, he asked me a few questions, but they weren't stupid "Have you tried turning the computer on?" questions.
And then he said, "Sounds like you have an overheating problem. We'll send you out a new motherboard."
I am pleased.