The unit shut down on it's own during boot up, likely because it reached an over-temperature limit condition.

I've tried booting the machine from AC and from battery after the unit cooled. I've also tried holding the on switch for 5 seconds, but there is no indication of power getting to the machine: no lights turn on, not even a flicker.

Any ideas?

Try removing the memory modules and power on with your battery alone or ac adapter. If you can swap a known good battery that can isolate things more. Is the AC adapter LED light solid by the way?

Good luck

If you have opened the machine - check the coils. Bad solder on one pad. There are about 4 of them and should be on the right side near DVD/Extra battery power connector. Check the top and bottom sides of the board. They are rather large by comparison to the rest of the parts on the board, and if you can lift one side of the coil with your fingernail, that is the one. This was just discovered last week by me, on several motherboards that I had in the pending stack.

CPU over heating? or do as suggested in post #2 :)

1 - it is not an overheating problem. (Overheating was a guess)
2 - If a unit is overheating, it should still take about 2min before shutdown even if the fan is not working at all.
3 - my previous suggestion is a known problem. And had been recognized on several models doing the same thing.

I have repaired over 500 laptops and recognized the pattern of manufacturing failures in Thinkpads, Gateway, HP and the other common brands.

Disassemble the unit and check the larger coils (indicated by L1, L4 or something) they will be grey and 4 sided. One side is lifted -no contact. When the current tries to go though it, it warms up, and the connection breaks. Check and see!