This is a known bug in the IOS Mail app. 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.
Why is this happening?
The Mail app is showing you the emails that got moved into your Inbox most "recently" instead of your "newest" emails. So, if you or a service like SaneBox 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.
Thus, your Inbox will appear to have a set of missing emails. Don't panic. The emails are really still in your Inbox. You can verify that by viewing your Inbox in your web mail interface. The problem is that IOS Apple Mail hasn't "fetched" those gap emails yet and therefore is not showing them in your view.
Most all 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. Or fetch them all, which avoids the problem as well.
We and others have begged Apple to fix this problematic bug type behavior to no avail.
How do I fix this?
First begin with updating to the newest version of IOS.
Then try these other four possible solutions:
1. Keep scrolling down passed the bottom of your Inbox view to force the app to load more mail. Newer versions of Mail app automatically retrieve the next series of email from your server each time you hit the bottom. Older versions of iOS Mail (earlier than IOS 8) showed a "Load More Messages" button at the bottom of the list. Click that button if it appears.
Repeatedly scrolling down (or clicking the "Load More Messages" button) will make the app fetch the next batch of messages. Apparent gaps will be magically healed as Mail app caches the entire Inbox. This seemingly endless trip to the bottom of your Inbox is tedious, but it really does set things straight. Your newest mail will rise to the top again once you've pulled all the way to the actual bottom of your Inbox. Sounds crazy but it works.
2. For certain less common email systems (some Exchange hosting configurations for example) it may help to check IOS Settings->Mail->Accounts (your account) and set "Mail Days to Sync" to "No Limit". This option isn't shown unless your host supports it.
3. Select and move any range of old emails showing out of sequence in your Inbox into any other email folder. You can do this with any email client and/or web mail interface.
4. Signup or re-signup for SaneBox and allow us to help. After analyzing your email history, we will know which email is worthy of your Inbox. Your oldest emails will be moved to a SaneArchive folder and your less important email will be moved to SaneLater. Moving the old stuff and unimportant stuff our of your Inbox will eliminate the problem.
Remember that if you ever cancel your SaneBox account, choose to keep your Sane folders instead of having the email in those folders moved back to your Inbox. Keeping your Sane folders will avoid this IOS issue all together. After your SaneBox account is canceled those Sane folders are yours to do with as you wish. Remember, canceling your SaneBox 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.
SaneBox Compatibility with iOS Mail.
SaneBox works with any email client, device or provider (with the exception of POP-only providers, and there are very few of those left). Learn more.
The whole point of SaneBox is to get unimportant emails out of your way so that you can deal with your important ones now. We take all of those unimportant emails and put them into SaneLater so that can either archive or delete them when you have a free moment. Learn more.