In some allocation problems, there is a priority ordering of agents that must be respected in the matching that is produced, in that an agent cannot have revealed to prefer the assignment given to another agent with lower priority. Otherwise, the affected agent could “contest” the assignment, often legally. Mechanisms that never produce assignments that can be contested are Incontestable Mechanisms. While serial dictatorship is the unique Incontestable direct mechanism, coarser message spaces allow for a wider space of mechanisms. We provide characterizations of incontestable mechanisms for different restrictions on message spaces, evaluate incentives induced by these mechanisms and apply these concepts to a real-life application: the Indian Civil Service allocation.