I am having a similar issue. My product is Samsung Galaxy Tab 4
The email stopped updating around the same time I turned off notifications for Clash of Clans (I had not played in about a year, and went to look at it one day, then after I closed the CoC app, the app was sending me notifications, so I went to that old game's settings and turn off it's notifications. I am 99% sure I turned off the notifications for CoC only.
Other recent activity from me is having added PokemonGo a few weeks ago, after that, I bought a mobile hot spot, which I switch between my home Wifi and my mobile hot spot repeadedly (whenever I go out to play PokemonGo).
The timing of my new activities on my tablet seems to tie in with turning off Notifications for Clash of Clans, but this is only a speculation. It could be an update with PokemonGo for all I know.
For sure, I am not getting my newest emails, causing me to miss a very important email.
The last email I received on my tablet was on 08/15/2016 at 4:03pm pacific.
According to my Inbox accessed on my Desktop PC, I have received 18 emails since then. I personally do not like to check my emails on my PC, and checking emails on my tablet has been my primary task on my tablet since I bought it.
I have tapped the refresh icon in the inbox, but it does not show me my newest email. The time on the top says 8:29am, before and after this refresh, but it is 7:am-something right now, I assume that maybe the time it was days ago when it stopped updating? I'm not too sure.
I am very unskilled when it comes to using tablets, still do not recognize some of the image icons, like settings or some other squiggly, or triangle stuff. When helping me, if you would please speak as if I have never used a tablet before, and do not recognize any of the icons. Thanks, it would help me a lot.
Edit to Update: Tablet is finally working right again!
I noticed throughout the day today, realizing that my time on my tablet had no updated since Sunday, and the app ntpSync was unable to connect to it's server earlier today (ntpSync was installed along with ClockSync, to help keep my tablet's time up to date so I could play PokemonGo). Moments before this edit, the time has finally updated on it's own, with 20 new emails waiting for me. Not sure what broke it, or what fixed it, but it is suddenly working again. I had a friend say something about local fires 'maybe' affected things, we can only speculate. I am happy it is fixed.