Update (sorta)

After I changed the thermal paste as you suggested, I was still getting the reboots but not as frequently as before. Usually it happened after the computer had been running for a bit and I tried to run multiple applications at the same time. When it did happen, I shut it down and let it sit for a few hours (overnight) before trying again. Seemed to be OK.

Took a week's vacation so didn't use the computer at all. Came back and it was working OK until last night. Had the computer off all day and started it up after I came home. Was only on for a few hours when I got the reboots. Tried quite a few times to start it up but it keep rebooting over and over, sometimes not even letting me get as far as the Windows login screen. Got a message that BIOS had failed so that memory settings had been reset to default and I needed to hit F4 to continue. Did all that but still got nowhere so I shut it down for the evening.

Woke up this morning and tried again but was still getting the reboots. Sometimes I could login sometimes I couldn't even get that far. Tried safe mode a few times and same thing happened. HDDs were pretty hot but CPU heat sink didn't seem too bad. Also, CPU fan pretty old (2005) but still appears to be working OK. Not sure why it keeps doing this.

I know that fans can sometimes go bad but can the same thing happen to heat sinks. Trying to figure out if it's worth it to change heat sink, fan or both. Think that might help?