Pressure relief valves (PRVs) are included as an essential element of many compressor piping systems in order to prevent overpressurization and also to minimize the loss of process gas during relief events. Failure of the valve to operate properly can result in excessive quantities of vented gas and/or catastrophic failure of the piping system. Several mechanisms for chatter and instability have been previously identified for spring-loaded relief valves, but pilot-operated relief valves are widely considered to be stable. In this paper, pilot-operated PRVs are shown to be susceptible to a dynamic instability under certain conditions where valve dynamics couple with upstream piping acoustics. This self-exciting instability can cause severe oscillations of the valve piston, damaging the valve seat, preventing resealing, and possibly causing damage to attached piping. Two case studies are presented, which show damaging unstable oscillations in a field installation and a blowdown rig, and a methodology is presented for modeling the instability by coupling a valve dynamic model with a 1D transient fluid dynamics simulation code. Modeling results are compared with measured stable and unstable operation in a blowdown rig to show that the modeling approach accurately predicts the observed behaviors.