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As large-scale training regimes have gained popularity, the use of pretrained models for downstream tasks has become common practice in machine learning. While pretraining has been shown to enhance the performance of models in practice, the transfer of robustness properties from pretraining to downstream tasks remains poorly understood. In this study, we demonstrate that the robustness of a linear predictor on downstream tasks can be constrained by the robustness of its underlying representation, regardless of the protocol used for pretraining. We prove (i) a bound on the loss that holds independent of any downstream task, as well as (ii) a criterion for robust classification in particular. We validate our theoretical results in practical applications, show how our results can be used for calibrating expectations of downstream robustness, and when our results are useful for optimal transfer learning. Taken together, our results offer an initial step towards characterizing the requirements of the representation function for reliable post-adaptation performance.