Renaming is a fundamental problem in distributed computing, in which a set of n processes need to pick unique names from a namespace of limited size. In this paper, we present the first early-deciding upper bounds for synchronous renaming, in which the running time adapts to the actual number of failures f in the execution. We show that, surprisingly, renaming can be solved in \emphconstant time if the number of failures f is limited to O(n√) , while for general f ≤ n − 1 renaming can always be solved in O( logf ) communication rounds. In the wait-free case, i.e. for f = n − 1, our upper bounds match the Ω( logn ) lower bound of Chaudhuri et al. [13].