PSA: The "battery optimization" feature in Android 6.0 Marshmallow may be causing your Gmail to not notify you of e-mails until it's way too late and other apps to not notify you about anything ever.
I first caught on to this when I stumbled upon a Reddit thread on /r/Android. Essentially, Android 6.0 added a feature called Doze which puts apps to sleep while the phone is locked/not in use, giving them only brief windows of opportunity to check for any new notifications before being put back to sleep. The frequency between these wake opportunities depends on how long the phone has been locked. It sounds like a good idea in theory, but the execution was done very poorly.
That thread was specifically about apps like Gmail and Inbox. Users were seeing issues where Gmail wouldn't notify them about new e-mails until several hours after the e-mails came in. This is because the Gmail app was getting "dozed" and was unable to check for any new messages (or receive push notifications, or whatever it does). The work-around is to disable the "battery optimization" for the Gmail app.
To do so:
Doze makes it sound like not optimizing apps will hurt your battery life. This may be true for poorly written apps, but it's pretty safe to allow Gmail to not be put to sleep. Before Android 6.0, Doze wasn't even a feature anyway and there weren't a lot of complaints about battery life. Personally, I just disabled battery optimization for a small handful of apps that I care more about, including Gmail and Hangouts.
In addition to Gmail, a few other apps I found to be completely broken when they're being "optimized":
Google Opinion Rewards: Get Google Play credit for answering surveys. I noticed I wasn't getting any surveys offered for several weeks, and figured the Doze feature was to blame. I disabled it on the Google Opinion Rewards last night, and got a survey this morning.
My theory is that this app only receives surveys if there is one available at the time the app asks for it. Since the app was checking very infrequently, it was missing all the surveys.
Tasker: I have a Tasker task set up to automatically connect to my VPN when my phone joins certain WiFi networks. I was noticing that it was failing to do this the majority of the time. I'd have to open Tasker and manually play the Connect VPN task and then it would sorta work, but sometimes it would fail to run the Disconnect VPN task when I left the WiFi network.
Disabling battery optimization for Tasker fixed this problem.
Tumblr: I was getting no notifications at all from this app, ever. Disabling battery optimization fixed this problem.
Reddit Sync Pro: I was getting no notifications of new messages on this app until I manually opened it. Again, disabling the optimization helped.
So if you have any apps that have been oddly quiet for the past several weeks or months and you're running Android 6.0, check if the "battery optimization" feature is to blame.