Nexus 6 Powered On Strangely
Since you have confirmed that the battery is just six months old, I am guessing that this is a different problem

Nexus and HTC phones seem to have an issue with battery reading showing incorrectly and have a method for battery reset logic. This method varies between various models and googling for "Nexus 6 battery reset logic" led me to this method suggested by Claire LP. The problems described in that thread are not the same as yours but if your problem is related to resetting logic, it may help:


  
  Power off your phone.
  Press and hold the Power and Volume Down buttons. This will put the phone in fastboot.
  Use the Volume buttons to scroll through the menu until you see "Bootloader logs"
  Press and hold the power key for a long time (7+ seconds) until the phone reboots.