I have a T61 (type 7658 CTO) that suddenly fails to boot up. Prior to this, it turned on normally and was left on for about an hour. Sometime during that hour the laptop shut itself off. There was still a charge in the battery and it was plugged in to a power supply. Shortly after starting up there was a popup message saying that a Lenovo update was available but this message was ignored and not acted upon.
The only LED indicator lights that now work are the one indicating that it is plugged into a power supply and the one that indicates the battery is in use or charging. If the power switch is pressed, both of these LEDs turn off. Nothing else happens and the only way to get these two LEDs back on is to remove the battery and power plug and then reconnect both of them. No other LEDs turn on at any time and the hard drive is not active.
Sound like a hardware problem. I don't know what graphic card you've got in your computer but if you have the Nvidia Quadro NVS 140M it could be dead. Nvidia had a lot of problems with these chips. As they part of the main board, if your Thinkpad is not under warranty is game over.
All the 7658 boards are Intel, not nVIDIA. Pull the main battery and unplug it. Press the power button several times, holding it for a full 5 seconds on the last time. (That will drain capacitors and completely reset) Put the battery back in; plug it in and tyry again.