We present a new abstraction to replace the t of n assumption used in designing fault-tolerant algorithms. This abstraction models dependent process failures yet it is as simple to use as the t of n assumption. To illustrate this abstraction, we consider Consensus for synchronous systems with both crash and arbitrary process failures. By considering failure correlations, we are able to reduce latency and enable the solution of Consensus for system configurations in which it is not possible when forced to use algorithms designed under the t of n assumption. We show that, in general, the number of rounds required in the worst case when assuming crash failures is different from the number of rounds required when assuming arbitrary failures. This is in contrast with the traditional result under the t of n assumption.