I have seen a very similar problem on somewhat different hardware (64 bit AMD), but in that case it was to do with the network interface. If I left it plugged in by cable it would run for at least as long as I wanted it to, but after about 20 mins with the cable unplugged it would lock up hard.
I thought it was due to the wireless card kicking in and that was the cause of the problem, however even after extensive messing around with compiling more recent drivers the problem persisted. The problem was eventually "fixed" in a more recent kernel, but even now unless I deactivate the ethernet interface, it still pretends to go on-line, then off-line every 15 to 20 minutes.
All of which is no real help as the problem of locking up doesn't occur with the kernel provided in Mint 13, but I'm prepared to bet that the problem is down to some hardware driver misbehaving, the trouble is finding out which one. If you had a proper installation you could look in the system log to see if anything odd is recorded just prior to each lock up. Otherwise, I suppose you could have a terminal open with
tail -f /var/log/syslog
running, but there’s no guarantee that it will show you what is going wrong before it goes TU.