We compare the solvability of the Consensus and Broadcast problems in synchronous communication networks in which the delivery of messages is not reliable. The failure model is the mobile omission faults model. During each round, some messages can be lost and the set of possible simultaneous losses is the same for each round. We investigate these problems for the first time for arbitrary sets of possible failures. Previously, these sets were defined by bounding the numbers of failures. In this setting, we present a new necessary condition for the solvability of Consensus that unifies previous impossibility results in this area. This condition is expressed using Broadcastability properties. As a very important application, we show that when the sets of omissions that can occur are defined by bounding the numbers of failures, counted in any way (locally, globally, etc.), then the Consensus problem is actually equivalent to the Broadcast problem.