If exchange account email push stopped working completely and non-exchange email account continued to push.
Also exchange account only update itself when you open the mailbox.
Fill all settings and make sure you will see tick marks.
Setup your exchange email account using Settings - Add Account.
You can in fact access the more recent items (via your web mail or even an alternate email app or email client.) The Mail app is actually grabbing the wrong subset of your Inbox email which causes some email to appear to be missing.
The Mail app is showing you the emails that got moved into your Inbox most "recently" instead of your "newest" emails.
Your newest mail will rise to the top again once you've pulled all the way to the actual bottom of your Inbox. Signup or re-signup for Sane Box and allow us to help.
After analyzing your email history, we will know which email is worthy of your Inbox.
Moving the old stuff and unimportant stuff our of your Inbox will eliminate the problem.
So, if you or a service like Sane Box copies a large number of older emails into your Inbox right now, you may only see those and brand new emails that arrived after the copy. You can verify that by viewing your Inbox in your web mail interface.
Thus, your Inbox will appear to have a set of missing emails. The problem is that IOS Apple Mail hasn't "fetched" those gap emails yet and therefore is not showing them in your other email clients on the face of the planet (including Apple OSX Mail) fetch from your mail server by latest arrivals first instead of most recently copied or moved.
After your Sane Box account is canceled those Sane folders are yours to do with as you wish.
Remember, canceling your Sane Box account causes us to securely erase our copy of your username/password or Gmail/Yahoo/AOL/Outlook authentication token so we wouldn't be able to continue to filter your Inbox email into those folders even if you wanted us to.