So far, the distributed computing community has either assumed that all the processes of a distributed system have distinct identifiers or, more rarely, that the processes are anonymous and have no identifiers. These are two extremes of the same general model: namely, n processes use l different authenticated identifiers, where 1 ≤ l ≤ n. In this paper, we ask how many identifiers are actually needed to reach agreement in a distributed system with t Byzantine processes. We show that having 3t+1 identifiers is necessary and sufficient for agreement in the synchronous case but, more surprisingly, the number of identifiers must be greater than n+3t/2 in the partially synchronous case. This demonstrates two differences from the classical model (which has l=n): there are situations where relaxing synchrony to partial synchrony renders agreement impossible; and, in the partially synchronous case, increasing the number of correct processes can actually make it harder to reach agreement. The impossibility proofs use the fact that a Byzantine process can send multiple messages to the same recipient in a round. We show that removing this ability makes agreement easier: then, t+1 identifiers are sufficient for agreement, even in the partially synchronous model.