First of all, my Mac Book Pro won't start up after going to sleep mode - all black screen etc. When I choose power down from the menu, well, it doesn't power down completely. The hard drive is still running. Only when I hold down the power button will it power down properly. When I try to boot, the Mac book simply won't turn on; nothing happens until I keep pushing and pushing several times then it turns on, but there is no pattern to how many times I have to push the button. The super strange thing is that it sometimes turns on after a good while, say anything from between 5 min - 30m - 1 hour AFTER I've pushed the power button. And the Mac Book prior to that has no sign of being in sleep mode or anything at all, no heat or anything - just cold and dead. Then I turn my back and then as if nothing has happened it boots up!? I've tried several routines like taking off the power cord, dropping the battery, then holding down the power button for several seconds before then applying the power cord and again putting in the battery, etc. Sometimes this has an effect, sometimes not. Sometimes it seems it helps to take off the powercord immediately after pushing the power button several times, but maybe it's just luck. It also seems that my right USB is recognising devices slower than before and sometimes not at all. My left usb is dead.
Once you've gotten your backup - get into Disk Utility and try to repair the disk / consider an erase install. It will likely fail (unless the delay was caused by a very unlikely file system corruption or startup conflict).
Dropping a clean install - only the OS will let you know right away if the Hard Drive (HDD) can take new data or if other hardware issues are causing the half hour delay. Something fairly destructive is needed to cause a mac to take that long to successfully boot. Take it to a hardware tech if you have any doubts how to troubleshoot this, but odds are it's storage related.