A further update: it seems to make a difference whether I have one DIMM installed or two.
The system originally came with just one 4GB DIMM. I've done the only upgrade possible on this system: added a second 4GB DIMM to take the total RAM to 8GB.
After upgrading to 8GB, I had run Memtest86+ for 11 hours, and Prime95, and no errors were reported. Also, Windows has not crashed or seemed unstable.
However, I've now done some further testing with Prime95, running the Blend test. Sometimes, after a few hours, it'll encounter this: "FATAL ERROR: Rounding was 0.5, expected less than 0.4". Checking the Windows system log in Event Viewer, at around the same time I see an error "Windows removed bad memory regions from this PC". It's weird that it only happens sometimes, and only after several hours of Prime95 testing have passed.
So perhaps AIDA64's System Stability Test is encountering a similar problem when 2 DIMMs are installed in this system, but it's doing so much more quickly. If so, surely it would be more helpful if there was an error message of some kind, instead of just stopping silently and then not doing anything?