Well... I am taking my Note 5 back to Verizon tomorrow for this exact reason - and spent HOURS on the phone the past two days with Verizon and Samsung tech reps to try to figure it out. My push emails simply never worked more than a couple of times.
I have home security cameras and when they detect movement outside the home, they send a fast push email to let you know. On my iPhone 6, I got those notifications within seconds of their sending them. No such luck with the Note 5. In another thread where people were discussing this same issue with the S6, I tried literally EVERY suggestion people were coming up with - nothing worked permanently and only a couple of things worked for maybe a few minutes to an hour, then the phone would slip right back into not getting them again. I think people were starting to lean toward something in the Samsung software that keeps disabling them or something, but no one has figured it out yet.
I literally made 4 calls to Verizon tech totaling 4+ hours on the phone in the past two days and was lucky enough to get knowledgeable reps each time. I kid you not - these guys literally picked apart every setting you could possibly imagine in Gmail, Android, etc. Then moved on to just removing gmail and replacing it. Nothing. Trying an alternative email other than gmail - nope. We tried Samsung tech support (India call center) and they could not fix it after two hours allegedly with a "level 2" rep. As a final resort we completely factory reset the phone and started from scratch... same issue. Pushes not working.
Sorry - enough is enough. I love everything else about this phone but between some work emails and the security system, I HAVE to have fast push notifications from email coming through to me, and it's just not going to happen with this phone. I'm irritated to be honest... loved just about everything else on this phone. But switching back to my iPhone 6 for now and guess I'll just shell out more money for the 6s+ when it comes out if I want a larger screen phone that works out of the box. UGH. Samsung... SLAP!!!! :-$