I'm having email sync issues & can't find a solution that fixes my problem. Tried everything I've been able to find, including ensuring Sync function is enabled in multiple locations, disabling then reenabling sync features, powering my phone off/on, even resetting the battery. Received the NK1 update on 11 Dec. Not sure if that's when the problems started or not, as I've been busy, but that's seems about the time things stopped sync'ing properly. Am using the email app that comes with the phone (an S4 mini), and have 3 email accounts, cox.net, and 2 aol accts. The cox & 1 of the aol accts won't sync, but the other aol acct does sync. The one that is working I have set to not auto sync, & the manual sync works. Not so on the other two ... the sync app says they are sync'd, but the email app says they haven't been sync'd in like 2-3 days, even tho I've sync'd them multiple times today. Get a continually circling busy / working symbol. Get new email on desktop computer without issues. After a few days, the other two accts will sync up.
From time-to-time I have had this issue with the default email app. Have you checked to see that the app is synched? When it happens with my S4, there is a message at the bottom of the screen saying the app isn't synched. Try this: pull down the status bar and click on the icon at the far right. This will open all the functions available on the status bar. Make sure 'Synch' is on. For some reason, this function on my phone gets turned off without me doing it. KarenC suggested using another email app. As a backup, I use K-9 Mail. I really like this app.