While using the latest version of the Gmail app: 6.3.117716127, I cannot sync my mail whatsoever.
Update: 6.4.120760496 has the same issue.
Please note that this is NOT a universal "Google" sign in problem, where you get a black background with white text and you have to enter your credentials, although the message is the same ("Couldn't sign in")
This is for the Gmail app only. Observe the error in my screenshot below:
Tapping "Sign in" just makes the message at the bottom go away, but nothing else happens, and it doesn't sync. If I try refreshing my inbox, it will bring that sign-in message back.
My last sync with the newest version was April 13th. Then I reverted back to an older version of Gmail from November of 2015. That version syncs just fine.
You can see email from April 19 in the screenshot, because I tried upgrading back to the latest version, but it still wouldn't sync.
Deleting the cache and data doesn't help.
I did not change my Google password.
My phone is a Galaxy S4 running Android 4.4.2 (Hyperdrive)