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We investigate experiments that are designed to select a treatment arm for population deployment. Multi-armed bandit algorithms can enhance efficiency by dynamically allocating measurement effort towards higher performing arms based on observed feedback. However, such dynamics can result in brittle behavior in the face of nonstationary exogenous factors influencing arms' performance during the experiment. To counter this, we propose deconfounded Thompson sampling (DTS), a more robust variant of the prominent Thompson sampling algorithm. As observations accumulate, DTS projects the population-level performance of an arm while controlling for the context within which observed treatment decisions were made. Contexts here might capture a comprehensible source of variation, such as the country of a treated individual, or simply record the time of treatment. We provide bounds on both within-experiment and post-experiment regret of DTS, illustrating its resilience to exogenous variation and the delicate balance it strikes between exploration and exploitation. Our proofs leverage inverse propensity weights to analyze the evolution of the posterior distribution, a departure from established methods in the literature. Hinting that new understanding is indeed necessary, we show that a deconfounded variant of the popular upper confidence bound algorithm can fail completely.