Russ' Do It Yourself Home Workshop

If you have Outlook hooked up to an Exchange mail box (Office 365 in this case) and mail stops dropping into your inbox, try this.

Right-click on the offending Inbox and click Properties…

Under the General tab, click Clear Offline Items.  This will empty the cache.  In a scary fashion, all your inbox mail will disappear and (should) will re-sync with Exchange and re-populate.

Hopefully this will clear up your issue.  It did mine.