I'm having this problem with Gmail when emails are not synced. When I go to Settings->Accounts, I can see sync error. When I sync manually, or access email on other device, they get synced ok.
I asked on Google support forum and tried all their suggestions - removing and re-adding Google account, changing number of days to sync etc, nothing worked. Eventually, the suggestion was to check if any apps with Google account access are causing the problem. I found that when I removed Dropbox, it solved it. Or so I thought. It still happens. The stock email app has no problems at all. Sometimes emails will not arrive at night - maybe I need to add Gmail under battery non-optimised apps?
I'm using Marshmallow on M4 Aqua with latest Gmail app.
Edit: I suspect that even this forum sign in through Google account might be causing this. I will remove access and see if it fixes it.
