There's a few places where things can go wrong, most of them are caught automatically and you as a user never notice but some might require manual intervention, either from your side or by us. We're trying to document all of these cases here.
No new mails appearing in the inbox
If you are able to send mail, and can find new emails appearing in your sent folder (and all others) but not your inbox it is most likely due to it growing too large and when we try to retrieve it, the server-to-server communication times out. The quicket fix is to archive a set of older emails. This problem usually only comes up with very large folders (> 10000 emails).