OK peeps --- I reported this issue 2 months ago under 5.0 .... per a lot of recommendations, I checked every darn setting and could NOT get Notifications PUSHED for my stock EMAIL or GMAIL app. I cleared Cache, uninstalled apps, tried other email app (K9) , ensure NOTIFICATIONS are turned on for every App, tried SAFE Mode -- nothing worked. So I was really hoping 5.1.1 would solve this issue for me. It did NOT. So I did the painful Factory Reset, didn't immediately reinstall ANY of my apps and tested it ... No Push Notifications for EMAIL/GMAIL at all ! what a total waste of time & pain in the *ss that was ... a full day of effort reconfiguring everything.
to be clear as to the problem: When my S6 goes into Sleep mode (times out), I go to my laptop and send myself an email ... and wait .. and wait .. an hour later I wake up my phone - no notification / no email at all ... but then suddenly, "Finding Location" pops up and DING ... email arrives. I even tried turning off WiFi, suspecting that had something to do with this.. No difference. I also turned off SMART NETWORK SWITCH. nada.
Do a GOOGLE search on "not getting PUSH notifications Galaxy S6" and you'll find that this is a wide-spread issue. (I'm happy for those of you that is DOES work!) This really REALLY sucks. I never had this issue on my previous MOTO RAZR. Samsung or Google needs to admit there is an issue and step up and fix this! And for the record, I am very computer & smartphone literate. If there is a setting hidden that affects this, I can't find it - I have been thru every setting and nothing affects this.