This paper proposes a valid bootstrap-based distributional approximation for M-estimators exhibiting a Chernoff (1964)-type limiting distribution. For estimators of this kind, the standard nonparametric bootstrap is inconsistent. The method proposed herein is based on the nonparametric bootstrap, but restores consistency by altering the shape of the criterion function defining the estimator whose distribution we seek to approximate. This modification leads to a generic and easy-to-implement resampling method for inference that is conceptually distinct from other available distributional approximations. We illustrate the applicability of our results with four examples in econometrics and machine learning.