I had this problem off and on and I somehow didn't put 2&2 together to figure out that it was an HP problem instead of more likely just a simple fluke caused by a power glitch. My problem was identical to the ones described here but being in IT for 35 years, I knew to remove the battery and power to test which is something needed often with other equipment here multiple power outages occurr. Such was the case and I thought it just a fluke and not the recurring problem it turned into.
First time this happened about 6 months into ownership. I did the battery and AC power off procedure and the computer then came back to life. Then it happened again twice when I was out of the country while visiting a friend. Then as I returned to the US and at 13 months now into ownership of my Probook 4350s, the battery shorted dead out. I called HP to try to get warranty, explaining the history of the power problems with this model for me. So I explained this to HP support, that they had a lemon here and asked them to waive the one month out of warranty situation.
After attempted escalation it was a solid no, I could pay $85 for a new battery and that would fix the problem they said. Right! I told them what to do with their $85 battery and instead I bought one off eBay.
Everything seems perfect now, so I am not sure if the design is such that the battery has some type of feedback mechanism that is driving everyone's 4350's crazy or what the problem is. All I do know is that my problem is fixed, because this is my first and last HP laptop. I'm 3 months into the new battery with no more "where's Waldo" or hot flashes when I hit the power button and nothing happens.
Good luck everyone with your battery and unable to power up issues!