If you are using a Nexus device on Android 6.0 or later, whether that be the new Nexus 6P, 5X or older Nexus devices like the Nexus 6 and Nexus 5, you may have had some syncing issues with both Gmail and Inbox. It appears to be a known issue that the Android team is supposedly working on. Essentially, what's going on is that notifications for Gmail, Inbox and a few other apps are not coming in when they should be. They are either coming in much later, or not at all. Theres also an issue with Google Now and less definable errors. Also occurring with other Google services.
The issue is in the AOSP issue tracker, and has over 140 people who starred the thread, and Google representatives have indeed commented on them. So the Android team is on the issue, and hopefully a solution for the issue is coming soon. Currently, doing things like resetting Google accounts, clearing the cache, factory reset and other things have not reliably fixed the issue with notifications not coming in on time.
A few users in the AOSP issue tracker thread have noted that this may be due to the Doze feature included in Android 6.0. Some users have noted that the issue has been solved by turning off Doze. But it has only worked for some and not everyone that has been having the issue. It's tough to say what exactly is the culprit, but it can get a bit annoying – especially if you rely on email coming in. It may be a good idea to try another email app in the mean time if you need your email as soon as possible.
There's no promise, but hopefully the Android team can get it fixed up and include the fix in Android 6.0.2 or even the next security update (speaking of which, that should be coming in the next few days, as it is now January). We'll keep an eye on this issue and let everyone know as soon as Google have fixed the issue. In the mean time, we'll just have to open Gmail or Inbox every so often to check our email.