We study implementations of basic fault-tolerant primitives, such as consensus and registers, in message-passing systems subject to process crashes and a broad range of communication failures. Our results characterize the necessary and sufficient conditions for implementing these primitives as a function of the connectivity constraints and synchrony assumptions. Our main contribution is a new algorithm for partially synchronous consensus that is resilient to process crashes and channel failures and is optimal in its connectivity requirements. In contrast to prior work, our algorithm assumes the most general model of message loss where faulty channels are flaky, i.e., can lose messages without any guarantee of fairness. This failure model is particularly challenging for consensus algorithms, as it rules out standard solutions based on leader oracles and failure detectors. To circumvent this limitation, we construct our solution using a new variant of the recently proposed view synchronizer abstraction, which we adapt to the crash-prone setting with flaky channels.