Something that others haven't mentioned so far: is the fan working? Can you hear it start and accelerate as the CPU gets hotter?
Install a sensor monitoring package.
lm-sensors provides the command-line
sensors command, but you must first run
sudo /usr/sbin/sensors-detect — GUI options exist too. Thinkpads have good ACPI support for their fan(s), or at least every one I've owned since 2004 has. One upside of this is that you'll also get CPU temperature gauges. My T61p reports the single fan speed, CPU core temperatures (via the internal sensor), mini PCI (i.e. WLAN card) temperature, mainboard temperatuer and GPU temperature.
If the fan doesn't start, you have your (first) culprit. You'll need to have it replaced. Doing it yourself is possible too, if the machine is out of warranty, but it's an annoying little operation. Do it on a good work surface with good lighting, get a service manual and follow the instructions to the letter. You don't want to compromise that part of the machine.
If the fan does start, can you feel the airflow as it reaches full speed? On many Thinkpads, the exhaust is on the left side, with intakes in the rear and/or underside.
Do you clean the cooling channel regularly? My boss and I have identical Thinkpads, but his used to run 10°C hotter. He's a smoker, and didn't clean the vents regularly. Dust is a great insulator, and even with your fan going full speed, the airflow won't remove enough heat from the heatsink metal. To clean the vents, I use one a heavy duty manual air blower like the ones photographers use (my personal favourite — cheaper than air in a can, friendlier to the environment and surprisingly good exercise). Put the nozzle on the air intake and blow repeatedly. If this is the first time you do this, do it outdoors and/or wear a mask. Insane amounts of dust will come out of the exhaust vent. On mine, plumes come out of the left speaker holes too.
Below are basically musings which probably don't apply to you since you can stop the issue by cooling the laptop and/or reducing the CPU speed. You should almost certainly concentrate on the heat aspect.
If the CPU, GPU etc temperatures are all nominal, the fan turns on freely and does its work, and the machine still crashes when working hard, there are other (sadder) diagnoses. The primary suspects, in decreasing order of likelihood, are: damaged memory, damaged CPU, damaged mainboard, or a PSU that isn't delivering enough power.
You can test the memory easily enough. Just boot into
memtest86+, which you can install on Ubuntu (it'll appear on your GRUB boot menu) and let it do its job. If it finds dead memory, you have two options: replace the memory or, if the damage is small, tell Linux not to use the damaged area. If you need to replace memory, make sure your Thinkpad has all its memory off-board. Some have all their RAM on SODIMM slots, others have an amount permanently installed on the mainboard. If your on-board RAM is damaged, you'll have to have the mainboard replaced.
Ditto if the CPU or mainboard are found to be damaged, but I don't know how to make sure. I've never had issues of this sort on my otherwise quite abused Thinkpads.