I've had this issue since updating to iOS 8.3 with both my gmail account and my U of M alumni address, which is also a google account as of a couple years ago. I tried deleting and re-adding the U of M account (which is the one I primarily use), got the messages saying that the process couldn't be completed using SSL and saying it couldn't be completed at all, and tried adding it anyway, but of course it still couldn't get mail so I deleted it again.
I've power cycled the device, which didn't help. I filled out the unlock form and deleted the gmail account, and now it won't complete the process of re-adding it. I filled out the unlock form again and tried adding the U of M account again and it still doesn't work.
Throughout all of this, Yahoo mail works fine (but that's my secondary back-up and almost none of my contacts has that address), I can access my Google and U of M mail using the using the browser, but I don't like the Google webmail interface and I particularly hate the mobile version.
How do I get my Google-based mail accounts to work with iOS 8.3?
Thanks.
Troy