Replanning is a powerful mechanism for controlling robot motion under hard constraints and unpredictable disturbances, but it involves an inherent tradeoff between the planner's power (e.g., a planning horizon or time cutoff) and its responsiveness to disturbances. We present a real-time replanning technique that uses adaptive time stepping to learn the amount of time needed for a sample-based motion planner to make monotonic progress toward the goal. The technique is robust to the typically high variance exhibited by planning queries, and we prove that it is asymptotically complete for a deterministic environment and a static objective. For unpredictable environments, we present an adaptive time stepping contingency planning algorithm that achieves simultaneous safety-seeking and goal-seeking motion. These techniques generate responsive and safe motion in simulated scenarios across a range of difficulties, including applications to pursuit-evasion and aggressive collision-free teleoperation of an industrial robot arm in a cluttered environment.