Given a coherent system, let Z be the age of the machine at breakdown, and I the set of parts failed by time Z. Assume that the component lifetimes are independent. Assume further that the lifetime distributions are mutually absolutely continuous and that each possesses a single positive atom at the common essential infimum. We prove that the joint distribution of (Z, I) identifies the lifetime distribution of each part if and only if there is at most one component belonging to all cut sets. If we relax the mutual absolute continuity assumption by allowing isolated intervals of constancy, then a necessary and sufficient condition for identifiability is that no two parts be in parallel.