I have my Nexus 9 (on 5.x) connected to my iCloud account for email. It works fine, I even get notifications for new email. It can connect to the server just fine.
However, I get notifications from Gmail around every 15-20 minutes saying "Couldn't sign in" with my iCloud email address.
I don't like my tablet and wrist vibrating that often with useless notifications. How can I turn these off (or, ideally, fix the problem) without turning off notifications for real email?