Vehicle-to-vehicle (V2V) communication is a crucial component of the future autonomous driving systems since it enables improved awareness of the surrounding environment, even without extensive processing of sensory information. However, V2V communication is prone to failures and delays, so a distributed fault-tolerant approach is required for safe and efficient transportation. In this paper, we focus on the intersection crossing (IC) problem with autonomous vehicles that cooperate via V2V communications, and propose a novel distributed IC algorithm that can handle an unknown number of communication failures. Our analysis shows that both safety and liveness requirements are satisfied in all realistic situations. We also found, based on a real data set, that the crossing delay is only slightly increased even in the presence of highly correlated failures.