I have a 2 months old W520 factory settings. It has been acting up lately when powering up but after holding down the power button for 10 sec or so it would always start. However, that is not even working today.
I took out the battery and plugged it in AC and tried to no avail. I also tried powering it up with battery alone and neither seems to work.
A few of our communtiy users found the following method works for them:
Unplug machine from Power Supply
Hold down power button for 20 seconds
Press the power button 5 times
Install battery by re-plug it in
Press power button to turn it on again
The process drains capacitors on the motherboard that might be charged