God this dratted thing has happened to me too...
I have had my 5660 for two years, and then pop screen goes black.
Exactly the same symptoms as you two. EXACTLY.
I'm in New Zealand so sending in my laptop was not an economically viable option at all.
But after carrying my laptop to a laptop specialist, having acquired the 5660 service manual (cheers bsmith), IT SUDDENLY STARTED UP!
WINDOWS START UP!
windows said expected shutdown error due to ATI Radeon 9000 driver, caught in an infinite loop.
so yes, I took it home, and surely enough, when I tried to start it up.
Sensing a loose connection on the MB, in my rage I started knocking the laptop in certain places.
SURE ENOUGH I GET IT TO START UP.
I managed to replace pretty much every driver, Radeon driver several times. Didn't update bios though. Before it died on me.
Now, through trial and error, I can get my laptop to work for about a week (turning off at night/starting up in morning) by ONE light tap. When my computer dies, I force shutdown with power button for 10secs. Unplug everything. Close the lid. And give it an ever so light BUT FIRM tap on my knee, right in the MIDDLE of the bottom of the laptop (halfway from each side, and halfway from the front/back). This works a charm, and I am currently posting this message on my Sager 5660 which hasn't screwed up for 3 days now.
I did a bit of research and the only component that is quite susceptible to shock damage is the HDD. But when the computer is off, the laptop HDD can take a considerable amount more shock than when operating (x10-100 more acceleration). SO I only ever give it the knock when its fully unpluged etc.
You may think this is all crude, but I have had this problem for about 3 months now. And it has not gone away. Since I got the service manual, I have taken this computer to pieces, every possibly piece, fully about 5 times now. All of my theories of hope crushed by no success. A laptop engineer in new zealand thought the power controls were the problem, so I disassembled everything, and took the DC/DC board to him, 2 hour drive away! He ran a few diagnostics on it, not too sure what - probably connection/voltage in-out testing, and he said there was nothing AT ALL wrong with the DC/DC board. I have done most of what mskerman has done as well in diagnostics. I am stumped.
But you guys should try my knock theory and see if that works. It will be interesting to see if it does. IF it does, my question is what would a knock do to correct something? Not that many moving parts on the mobo, other than the actual components.
I really hate this problem, I have never failed in fixing a computer, I have 5 computers starting with a P1 133Mhz, and I have had a lot of problems in the 12 years of computing, both software and hardware, but this takes the cake. I have no idea of a permanent fix, but i have my own temporary fix that works but brings a tear to my eye when my computer blanks out and I lose whatever I was working on.
With shipping costs, and the price for a new mobo/upgrade mobo, I could buy a new laptop in NZ or Australia.
Dunno what to do?