Everyone....I found the real solution. My phone started doing the EXACT same thing. I was furious. It clearly wasn't a hardware issue because I used numerous microUSB cables and OEM USB 3.0 cables to no avail. It is clearly software related (unless other charges are working for you). Clearing the system cache took 30 seconds and my phone is back to normal without having to replace or lose any data. Clearing the system cache won't do anything to your phone that you will notice (contacts, apps, pictures, etc. are all still there). Here's how to do it:
Turn off the Galaxy Note 3.
Press and hold Volume UP + Home + Power buttons.
When the Android System Recovery screen appears, release the Volume Up and Home keys.
In the recovery menu, navigate to ‘’wipe cache partition’’ with the volume buttons.
Press power button to confirm.
Once this is done, select reboot system now.
Turn off the Galaxy Note 3.
Press and hold Volume UP + Home + Power buttons.
When the Android System Recovery screen appears, release the Volume Up and Home keys.
In the recovery menu, navigate to ‘’wipe cache partition’’ with the volume buttons.
Press power button to confirm.
Once this is done, select reboot system now.