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Tasks where robots must anticipate human intent, such as navigating around a cluttered home or sorting everyday items, are challenging because they exhibit a wide range of valid actions that lead to similar outcomes. Moreover, zero-shot cooperation between human-robot partners is an especially challenging problem because it requires the robot to infer and adapt on the fly to a latent human intent, which could vary significantly from human to human. Recently, deep learned motion prediction models have shown promising results in predicting human intent but are prone to being confidently incorrect. In this work, we present Risk-Calibrated Interactive Planning (RCIP), which is a framework for measuring and calibrating risk associated with uncertain action selection in human-robot cooperation, with the fundamental idea that the robot should ask for human clarification when the risk associated with the uncertainty in the human's intent cannot be controlled. RCIP builds on the theory of set-valued risk calibration to provide a finite-sample statistical guarantee on the cumulative loss incurred by the robot while minimizing the cost of human clarification in complex multi-step settings. Our main insight is to frame the risk control problem as a sequence-level multi-hypothesis testing problem, allowing efficient calibration using a low-dimensional parameter that controls a pre-trained risk-aware policy. Experiments across a variety of simulated and real-world environments demonstrate RCIP's ability to predict and adapt to a diverse set of dynamic human intents., Comment: Website with additional information, videos, and code: https://risk-calibrated-planning.github.io/