The behavior under particle loss of entanglement and nonlocality is investigated in multipartite quantum systems. In particular, we define a notion of persistency of nonlocality, which leads to device-independent tests of persistent entanglement. We investigate the persistency of various classes of multipartite quantum states, exhibiting a variety of different behaviors. A particular attention is devoted to states featuring maximal persistency. Finally we discuss a link between the symmetry of a state and its persistency, illustrating the fact that too much symmetry reduces the strength of correlations among subsystems. These ideas also lead to a device-independent estimation of the asymmetry of a quantum state.