A common chromebook issue that I see is that the chromebook will not turn on even though it has been charging. This is usually caused by a hardware error between the battery and the motherboard of the chromebook. An easy way to possibly fix the issue is to perform a hardware reset on the chromebook. This will erase all hardware properties saved in the chromebook and force it to "relearn" all the drivers for hardware connected to the chromebook. This will usually fix a "no power on" issue. If it does not, the battery may have to be taken out and replaced. The chromebook should power on like normal if the fix takes hold. 

Reset your Chromebook hardware
  1. Turn off your Chromebook.
  2. Press and hold Refresh + tap Power .
  3. When your Chromebook starts up, release Refresh .