If you've been experiencing some intermittent problems getting timely email alerts on your Nexus phone, you're not alone. According to lengthy threads on both Google's official Nexus Help Forum and the AOSP issue tracker, quite a few Android users running Android 6.0 and later builds on Nexus hardware are seeing similar issues. The problem seems to result in late or missing notifications for Gmail and Inbox, as well as less frequent alerts for other apps, and less definable errors with some Google services like Google Now.
The issue tracker entry has been starred by 140 people at the time of writing, indicating a fairly widespread problem, though it isn't universal. Various esoteric solutions like wiping the cache, resetting Google accounts, or flashing back to earlier Android builds and then applying the update don't seem to reliably solve the issues, and they're affecting a wide variety of Nexus phones and tablets. The first post was over six weeks ago, though the issue seems to be affecting a wider variety of users as it continues.
Google representatives have replied to both threads, so the Android team is almost certainly aware of the problem at this point. Various users have posited that the issue stems from Android 6.0's power-saving Doze feature, and turning it off helps some (but not all) end the problem. Hopefully it will all get sorted out in the next incremental update.