UPDATE 7/18/2016 12:10 EDT: If you’re are still experiencing the dreaded Pokemon Go “Authentication Error” you may have become aware that the majority of people affected by this problem have Pokemon Trainer Club accounts. So, if you’re still looking for a solution, here’s a few that you can try right now.
1. Some users have suggested that when they first came across the issue, they thought that their accounts had somehow been banned. However, the problem is not that at all! It appears that due to the huge success of the game that servers in different countries around the world have not been able to cope with demand. And as such this server problem could be the reason for your authentication problem.
You can check to see if your local servers are up or down here, however from what we can tell this problem does not affect Pokemon Go users logged in via their Google accounts, so if you are effected keep checking and attempt to log on when they’re up again.