There is great interest in methods to improve human insight into trained non-linear models. Leading approaches include producing a ranking of the most relevant features, a non-trivial task for non-linear models. We show theoretically and empirically the benefit of a novel version of recursive feature elimination (RFE) as often used with SVMs; the key idea is a simple twist on the kinds of sensitivity testing employed in computational learning theory with membership queries (e.g., [1]). With membership queries, one can check whether changing the value of a feature in an example changes the label. In the real-world, we usually cannot get answers to such queries, so our approach instead makes these queries to a trained (imperfect) non-linear model. Because SVMs are widely used in bioinformatics, our empirical results use a real-world cancer genomics problem; because ground truth is not known for this task, we discuss the potential insights provided. We also evaluate on synthetic data where ground truth is known.