Robust synchronization is essential to ensure the stable operation of many complex networked systems such as electric power grids. Increasing energy demands and more strongly distributing power sources raise the question of where to add new connection lines to the already existing grid. Here we study how the addition of individual links impacts the emergence of synchrony in oscillator networks that model power grids on coarse scales. We reveal that adding new links may not only promote but also destroy synchrony and link this counter-intuitive phenomenon to Braess's paradox known for traffic networks. We analytically uncover its underlying mechanism in an elementary grid example, trace its origin to geometric frustration in phase oscillators, and show that it generically occurs across a wide range of systems. As an important consequence, upgrading the grid requires particular care when adding new connections because some may destabilize the synchronization of the grid - and thus induce power outages.