I have seen and experienced emails dissappearing or not being able to receive new items or sent emails not being saved in sent items only when rules get corrupted.
I normally then start outlook without rules and this has fixed for me.
If you are not the exchange admin, then consult your exchange admin as they might have some sort of policy going on which can cause this, from an archiving rule to a folder policy (MRM).
Also make sure you haven't got outlook open anywhere else.