Issue: Chromebook is asking for an old password(see screenshot below for example).
Resolution: Unless there was something important stored locally, you can bypass this and refresh all your information from your google account. Just click on "Forgot your old password?" and click on the PROCEED ANYWAY button. (See screenshot below).
Additional information: This is caused by a user logging into a chromebook, then changing their password then trying to log into the same chromebook again. Chrome recognizes the password discrepancy and attempts to verify.