What mail app are you using? Have you tried another app? As for the app you are using and why it won't sync... I can't think of a reason that it wouldn't sync if you have internet and basically have requested a forced sync by manually refreshing the account, unless there was an update to the app or there was an update that corrupted the app. You've deleted the account and added it back, did any of your old emails return after that? If it's a third party app you could try uninstalling it and install it again and then add your accounts back.... If it's a stock app you could maybe get the same results by uninstalling the updates > then disabling the app > restarting the device > enable the app > then installing the updates and your accounts... Just thinking out loud you have internet, for you to add an account it would have to make contact with the server to verify your email and password, which should have refreshed your emails at the same time...... The only other thing I can think of is something that shouldn't have changed if it was working before which is an incorrect server name or port # being used for your inbound items. When it verifies your email and password I believe it only checks your outbound sever and port because I want to say that I've had an account that I could send items but not receive them due to a typo on the inbound sever name. Like I said that shouldn't have changed if it was working unless they (your email provider) changed their protocols. Which should have sparked a lot more posts about email apps not updating... Try the uninstall or disable, if neither of them work I'd try a new app. I'd try the new app for two reasons One to see if it works which could point to a problem with the old app and Two if it still doesn't work it could point to some issue with your device, but what would make a device stop updating email I can't say.???
One last thing came to mind, have you disabled anything in your apps recently? I once disabled some things I thought were useless only to realize two or three days later that I wasn't able to make calls. I could receive them but I couldn't make any outbound calls due to disabling something that affected the touch pad.