Autonomous mobile robots require predictions of human motion to plan a safe trajectory that avoids them. Because human motion cannot be predicted exactly, future trajectories are typically inferred from real-world data via learning-based approximations. These approximations provide useful information on the pedestrian's behavior, but may deviate from the data, which can lead to collisions during planning. In this work, we introduce a joint prediction and planning framework, Partitioned Scenario Replay (PSR), that stores and partitions previously observed human trajectories, referred to as scenarios. During planning, scenarios observed in similar situations are reintroduced (or replayed) as motion predictions. By sampling real data and by building on scenario optimization and predictive control, the planner provides probabilistic collision avoidance guarantees in the real-world. Relying on this guarantee to remain safe, PSR can incrementally improve its prediction and planning performance online. We demonstrate our approach on a mobile robot navigating around pedestrians.