Machine learning models are often used to decide who receives a loan, a job interview, or a public benefit. Models in such settings use features without considering their actionability. As a result, they can assign predictions that are \emph{fixed} -- meaning that individuals who are denied loans and interviews are, in fact, precluded from access to credit and employment. In this work, we introduce a procedure called recourse verification to test if a model assigns fixed predictions to its decision subjects. We propose a model-agnostic approach for verification with reachable sets -- i.e., the set of all points that a person can reach through their actions in feature space. We develop methods to construct reachable sets for discrete feature spaces, which can certify the responsiveness of any model by simply querying its predictions. We conduct a comprehensive empirical study on the infeasibility of recourse on datasets from consumer finance. Our results highlight how models can inadvertently preclude access by assigning fixed predictions and underscore the need to account for actionability in model development.