I've delt with this a few times. First try a soft reset. Take the battery out and hold the power button down for 1 minute with the battery out. Put the battery back in and try to power it on. If that doesn't work try hooking it up to your computer with a USB cord and see if your comp can see the phone and turns it on. Last if those 2 fail you can try to start it in safe mode.
Turn off the phone completely.
Press and hold the Power key and the Volume Down key.
When the phone starts to boot, release the Power key but continue holding the Vol Down key until the phone finished restarting.
Safe mode will be displayed at the lower left corner; you may release the Volume Down key now.
If safe mode doesn't work the last thing you can try is recovery mode.
This procedure will attempt to boot the phone up to its barebones. It means that the device will be booted up without loading the usual Android GUI. This is the final step so if it failed, then there’s nothing much you can do about the device but to let the technician sort things out for you.
You don’t need to do anything if the phone booted up in recovery mode successfully. All we wanted to know if it still capable of powering all its components up. Here’s how you boot in recovery…
Turn off the Galaxy Note 4 completely.
Press and hold the Volume Up and the Home keys together, then press and hold the Power key.
When the Note 4 vibrates, release both the Home and Power keys but continue holding the Volume Up key.
When the Android System Recovery shows on the screen, release the Vol Up key.
If it doesn't turn on in any of these steps you might need a new phone. The number 1 thing is to be 100% sure you have a good charged battery. If you can try it in a diff phone to make sure then do that. I had to buy a wall charger for mine and when it was charged it would turn green. My charge port was messed up and it would tell me I had 100% charge and then the phone would turn off and the charge port would stop working for days a time sometimes. Good luck