Laptop sometimes turns on sometimes not

I have a HP/Compaq nc6000 laptop that is a few years old running Windows XP. All service packs, etc are up to date.

Sometimes the laptop boots up normally and works fine and sometimes it doesn't. After pressing the power button, the power light comes on, the fan starts, (I think I hear) the hard drive spin and then it all stops. The power light goes to amber (charging) and then it turns off.

Typically when it boots properly, at the very beginning, the number lock, caps lock, and a third light all blink together, then turn off and it boots. When it's not going to boot, these lights don't even come on. (That's when I know it won't work.)

I've tried re-seating the memory. That worked twice but now I think it was just luck.

The first time I opened it, I also pulled a huge dust bunny from the fan. The rest of the insides were pretty clean and dust free. I blew everything out anyway.

Before this started happening, the machine was having some weirdness waking up after hibernating.

The problem happens sometimes after I shut it down properly and sometimes not. It's also booted up fine after a crash.

I've tried setting all the power controls to everything always on and leaving the computer on. But sometime in the middle of the night, it powers down. (I don't know whether this is a gentle power down or if it just shuts off instantly.)

The last time, it wouldn't start for a couple days and then I tried again and it just worked. As if nothing ever happened. I backed everything up on my external hard drive. And now it won't work again.

Any ideas?