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.