My Android 5.1 phone has a All/Priority/No notifications mechanism... I hold down the volume down button for a couple of seconds and I have a quick panel to choose which mode and duration selections for None/Priority. Requires going through Settings | Sounds | Interruptions and setting priority status per app, minor overhead.
I get call rings, txt message notifications, and almost nothing else until the Priority only mode expires.
I.e. I want to be able to check a list of not-important but informative notifications once or twice a day, and a couple of apps I want it to buzz and let me know immediately.
As it stands right now, I allow more apps to buzz and get my attention with their notifications than I'd prefer, all because I don't want to miss that information entirely.
I am getting a Pebble2 watch soon, though, so I'm hoping I can use that as my 'priority' notification filter, and turn off the buzzing on my phone.