Reaching consensus is fundamental for many coordination problems in distributed computing. Yet, this problem is impossible in an asynchronous system as soon as one process may crash. We show that an additional constraint on failure occurrences allows solving the consensus problem in crash-prone environments. We will consider so-called λ-constrained failures that can happen only while a predefined contention threshold λ has not yet been bypassed. Assuming these failures, we will present algorithms solving agreement problems and circumventing well-known impossibility results.