Considering that a CMOS reset seemed to have cleared up the problem at least for a while, the fault could definitely be on the motherboard itself.
1. Go through each of the steps in my last post, and post as much of a detailed answer to each suggestion as possible.
2. Give us the make/model/appoximate age of the computer.
3. You might have a faulty RAM module. If you have more than one stick of RAM installed, try running the system with only one of them installed at a time; see if the symptoms change depending on which particular RAM module is being used.
4. Replace the CMOS/BIOS battery on the motherboard. It probably won't fix the problem, but the batteries only cost 2 or 3 dollars, so it certainly can't hurt.