Problem I had:
Macbook Pro 17" from 2008. After a Lion update I would push the power button and get a solid white light (from the sleep light), black screen and there would be no chime. I had a disc in so I would hear it spinning.
I thought it was because of the Lion update.
The Macbook must have overheated when I was doing the Lion update. I tired SMC resets, taking the RAM out, PRAM zap, etc. I heard that this could be due to the Left Logic Board.
What I did was set the Macbook up like a teepee under the covers on the bed. I put lots of pillows around and covered it good. After it got really hot to the touch, I set it in another room upside down overnight. The next day it worked! I had the HDD out, so I powered it down and put it back in.
Problem was, I decided to do updates. About a year+ worth since that's how long it was down. So there was a 2GB combined Lion update. I did the update, restarted and the issue was back: solid white light, black screen, no chime.
I set it up like a teepee again under the covers and overheat it. That did not work. So a day later I powered it on, but this time I put it upside down on the bed with the lid closed, so the batter was facing up. With it powered on, I covered it really good. After about an hour give or take, I put it in the other room to let it cool. Once it was cool to the touch I powered it on and it worked.
I thought there was a correlation with Lion, but I think it is just a heat issue. While doing updates, the computer heats up.
So far it is still working. I downloaded smcfancontrol and set the fan speed to max. I am going to order a big cooling tray for it.
But it seems it was overheating and affecting some connectors on the logic board.
So in summary, the fix (at least for me) for the black screen, no chime, solid white light: is to overheat the laptop upside down under the bed covers and then let it set until it is cool. Then give it a shot. While I overheated mine, I took out the battery and HDD. Then download smcfancontrol.