The computer had some unusual problems, so a memory test was an obvious test.
I ran my usual memory test: memtest86
But when I ran the test, the computer would immediately reset.
After running a different memory test (with no fault found), I eventually realised that the fault was not with the RAM, it was with memtest86.
After some research, I found that running a 32-bit application (like memtest86), meant that there is scope for potential problems, as 32-bit applications can only address 4Gb, so at the limits of RAM, its possible for errors to cause problems.
So from now on, I’ll be careful about testing Ram of 4Gb or more.