Two settings are traditionally considered for secure multiparty computation, depending on whether or not a majority of the parties are assumed to be honest. Protocols designed under this assumption provide ``full security'' (and, in particular, guarantee output delivery and fairness) when this assumption holds; unfortunately, these protocols are completely insecure if this assumption is violated. On the other hand, protocols tolerating an arbitrary number of corruptions do not guarantee fairness or output delivery even if only a single party is dishonest. It is natural to wonder whether it is possible to achieve the ``best of both worlds'': namely, a single protocol that simultaneously achieves the best possible security in both the above settings. Here, we rule out this possibility (at least for general functionalities) but show some positive results regarding what can be achieved.