This is a very frustrating situation - because after an hour of fiddling, you push the button and it may just turn on, like nothing ever happened. One time, for us - it was days later. We feared the system was nearing the end of the road until we heard mention that it may be static causing the problem. Putting two and two together, this seemed to make sense as we'd get the occasional zap off the case. On further investigation, we found out it may have been a buildup of static in the motherboard which prevented the system from booting.
You're using the laptop when it switches off on you. Pressing the power button does not start it again. Your laptop will not boot, the power light comes on and that's about it. If you listen carefully, you may detect the lack of hard drive spinning. The hard disk light will probably not be lit.