Have you tried relogging in to Pokemon Go? If you are given the message "login failed" when in the loading screen, just hit "try a different account" and then relog in with your username and password.
If you are given that message when you log in with your username and password, although this isn't very scientific, my method is just to get off the game for a couple hours or days and it will always eventually let you log in, as long as you are using your correct account details.