Hibernate saves your state to the hard drive, some machines stay in a semi live state watching the keyboard(my desktop does) others power fully off and need the power button pressed again to boot then they load it back off the hard drive(my laptop does this)
If you have pressed the power button and it still wont start up, unplug it for 5 minutes, make yourself a sandwich, plug it back in, and see if it will start up, if not you managed to suffer a major component failure.