What percentage does the phone reach when you try to turn it on?
Some phones require the battery to be at a certain level before you can power them on. Other times, the phone turns on, but the power used during the boot up depletes the battery again which causes it to shut off.
This happens more frequently if you're not charging the phone with the factory included charger (leading to slower charging). And if your device supports quick charge 2.0+ and you have a compatable charger, you can probably avoid this issue altogether.
Just a guess though... could be somethihng else.