Over 150 messages about this problem have been received on the Issue Tracker site. A typical entry is one from a Nexus 6 owner who complains that if he leaves his phone idle for 30 minutes and then tries to send a test email to his Gmail account, he won't receive a notification. That would seem to indicate that the problem has something to with Doze, the new battery management tool that comes with Android 6.0. With Doze, the phone goes into a deep sleep after a period of inactivity, shutting down apps and disconnecting them from the network temporarily.
Trying the usual fixes like wiping the cache, performing a factory reset or flashing an earlier Android build, don't seem to help. Since Google employees have been responding to the messages on the Issue Tracker site, it seems to be a safe bet to say that Google is aware of the problem. Hopefully, a fix will soon be sent out.
Google Nexus 6P
source: IssueTracker via AndroidPolice