It is possible that the laptop overheated and that is why it shut down originally. And if it fried something it may well not start again until repaired. Or, if you are continually trying to restart it, it may not have cooled down sufficiently to actually allow it to turn on.
Once you get it to start up again, I would suggest you install and run monitoring software, there are quite a few free ones out there, that can monitor your temps and let you know if it is in fact over heating.
If it is still under warranty, you may want to contact the manufacturer.