How a treatment causes a particular outcome is a focus of inquiry in political science. When treatment data are either nonrandomly assigned or missing, the analyst will often invoke ignorability assumptions: that is, both the treatment and missingness are assumed to be as if randomly assigned, perhaps conditional on a set of observed covariates. But what if these assumptions are wrong? What if the analyst does not know why—or even if—a particular subject received a treatment? Building on Manski, Molinari offers an approach for calculating nonparametric identification bounds for the average treatment effect of a binary treatment under general missingness or nonrandom assignment. To make these bounds substantively more informative, Molinari's technique permits adding monotonicity assumptions (e.g., assuming that treatment effects are weakly positive). Given the potential importance of these assumptions, we develop a new Bayesian method for performing sensitivity analysis regarding them. This sensitivity analysis allows analysts to interpret the assumptions' consequences quantitatively and visually. We apply this method to two problems in political science, highlighting the method's utility for applied research.