As bandit algorithms are increasingly utilized in scientific studies and industrial applications, there is an associated increasing need for reliable inference methods based on the resulting adaptively-collected data. In this work, we develop methods for inference on data collected in batches using a bandit algorithm. We prove that the bandit arm selection probabilities cannot generally be assumed to concentrate. Non-concentration of the arm selection probabilities makes inference on adaptively-collected data challenging because classical statistical inference approaches, such as using asymptotic normality or the bootstrap, can have inflated Type-1 error and confidence intervals with below-nominal coverage probabilities even asymptotically. In response we develop the Batched Ordinary Least Squares estimator (BOLS) that we prove is (1) asymptotically normal on data collected from both multi-arm and contextual bandits and (2) robust to non-stationarity in the baseline reward and thus leads to reliable Type-1 error control and accurate confidence intervals.