If this happens when playing video, it sounds like high temps were detected in your gpu and the system went into standby to prevent damage.
It happened one time while watching video, but most of the time I was doing something else.
Monitor your temps with something like thinkpad fan control, your cpu should idle at 60-70 and gpu at 50-60. If one gets up around 90ish, your system will shut down (self protection).
I use Everest Ultimate in Windows which shows CPU temperature in my system tray. Idle is between 45-60, but stressed it quickly jumps to 75-85. I never have seen it hit the 90 degrees. I have Intel X3100 graphics which are not visible with Everest Ultimate.
It would also be a good idea to clean and reapply some artic silver #5, but be careful not to much, and make sure you have good contact on GPU and CPU.
I completely reassembled my laptop about 2 months ago (took it completely apart), and at that time I applied some artic silver #5. After the reassembly, my shutdown problems were gone
Unfortunately they returned when I installed the Intel 6300 ABGN network card last month
I've also seen a bad system board cause this problem. Take a good look at the bottom docking connector, if there is corrosion (one of many things that can cause this), you may have a bad board. On the one I had, I attempted to repair it, it even worked for a while... almost a year, but I'm not going to officially recommend anyone do this, but if you have nothing to lose, then you can give it a try.
Do this at your own risk, and only as last resort if your board has corrosion.
Mix distilled water and baking soda (you only need a few drops so you don't have to buy it, get an inverted cup held over a tea kettle at full boil and catch condensation and let it drip into a small container). Mix up paste and get an old toothbrush and thoroughly clean the connector, then thoroughly clean all the baking soda off the board and dry with a heatgun or hair dryer (to be safe let it set overnight just to be sure it's fully dry).
***Note: This worked for me, but I'm not making any statement on what effects baking soda may have on the electrical components.
Other then that, I can only suggest a new system board, but you might want to try removing/replacing as many components as possible, it's possible (but unlikely) that something is drawing to much current and forcing shutdown.
These may be long-shots, but it's all I can think of without seeing the machine.
Hmm, I see no signs of corrosion, but I will reassemble the laptop again, clean everything and see what happens. edit
: If that doesn't help, I will replace the motherboard. Unfortunately T61 4:3 boards with Intel Graphics are very hard to find, and if you do, they cost around $400... which is insane in my opinion.
T60 motherboards with Intel graphics are much cheaper, I can get one for approximate $50, but then I will need a T7x00 CPU as well.
But still, that will be cheaper than only a T61 motherboard.