Well ... maybe it's progress, sort of.
I bought two new 256MB DIMMs to stick in the machine. Having found the manual for the motherboard, I discovered that I've got 3 DIMM slots, and I can mix-and-match any combination of those slots I like, as long as I don't put more than 256MB in one slot.
To make a long story short ... I think I can avoid the crash problem by:
a) Taking the existing 128MB DIMMs out of the two slots they were in, and
b) Putting a new 256MB DIMM into the third slot (i.e. the slot that was empty).
I guess that gets me back to a stable condition with the existing machine, and it leaves it in the same basic configuration it was in before. But it probably means that I've got motherboard problems that will eventually rise up and kill the machine. Guess it's time to look at ordering a new desktop unit. (Sigh. I hate moving files and application programs between machines.)