Iphone randomly stops pushing email

Question

My iphone 5s has recently decided to stop pushing email...sometimes.  Today for example all was good until about the afternoon when I didn't get any more email messages, or so I thought, when I checked my computer tonight I had 5 new email messages but my phone showed nothing, it always fixes itself when I open up mail on my phone it gets the new messages then starts to push messages again...until randomly a few days later it stops.
This happens both with the mail app and my gmail app.  I can't seem to figure out it's problem.  I sometimes turn my wifi off and on when I don't want it to connect at work, but other than that there seems to be no logical reason for this.
All apps and software are up to date.  This only seems to have started happening about a month or two ago.
Other things like imessage and texts messages push just fine.

Answer 1

Rachel,
This reminds of me an issue we used to have with iPods -- often times it turns out to be a problem with one or more songs in your library. Were these all songs you purchased from the iTunes store, or did you get them elsewhere?
If elsewhere, you may need to try a rebuild of your iTunes library. Here's the easiest way to start (and to see if this is the problem). First of all, do not do anything without a backup of your music (you need a backup of your music anyway). If you don't already have one use DVD writable media and any backup software and make sure you've written all your songs out. This is particularly important on any songs you've purchased (if they came from CDs you own that can be considered your backup, although you will have to rip them again).
Now go to iTunes and in the library select all your songs (Ctrl-A) and press delete. Now -- be careful here, do not answer "Yes" when it asks you if you want to remove them from your hard drive. They will then disappear from iTunes but still be on your hard drive. Locate them on your hard drive (in Windows use Explorer). They will most likely be in your "My Music" folder but you can also search for them (if they came from outside the iTunes Music store they most likely have the extension MP3). Once you find them do not drag them all over into the library -- here's where we want to test something. Just drag over a few, not more than 10 or 20 (you can drag over a folder or two if each folder contains an album, which is normal).
Try syncing again and if that works then you know a couple of things. After that you can drag all your music back into your library and try a sync and if that doesn't work you know there is something wrong with one or more of your songs. But often this may clear up any problems.