Measurements of quantum systems extract only classical information. The resulting dichotomy between physically real and physically discoverable information underlies many of the most interesting phenomena of quantum theory. One such phenomenon is the capacity of quantum systems to encode information globally such that exhaustive measurement of their parts might not reveal it; this is the problem of local distinguishability. Here we focus on this problem and study simple qubit-based multipartite systems. We prove three results. First, arbitrarily large sets of arbitrarily multipartite states can always be found such that the states are completely mutually nonorthogonal, yet measurements on just one copy suffice to reduce the set of possible states of a system to two. Second, all locally indistinguishable sets of quantum states can be rendered perfectly distinguishable by the addition of just one system in a complementary set of purely nonorthogonal states. Third, sets of orthogonal product states exist that are perfectly LOCC distinguishable, yet only the completely nonorthogonal subsystem must always be measured during successful LOCC protocols.