Dynamic learning systems subject to selective labeling exhibit censoring, i.e. persistent negative predictions assigned to one or more subgroups of points. In applications like consumer finance, this results in groups of applicants that are persistently denied and thus never enter into the training data. In this work, we formalize censoring, demonstrate how it can arise, and highlight difficulties in detection. We consider safeguards against censoring - recourse and randomized-exploration - both of which ensure we collect labels for points that would otherwise go unobserved. The resulting techniques allow examples from censored groups to enter into the training data and correct the model. Our results highlight the otherwise unmeasured harms of censoring and demonstrate the effectiveness of mitigation strategies across a range of data generating processes.