In observational studies, identification of ATEs is generally achieved by assuming “no unmeasured confounding,” possibly after conditioning on enough covariates. Because this assumption is both strong and untestable, a sensitivity analysis should be performed. Common approaches include modeling the bias directly or varying the propensity scores to probe the effects of a potential unmeasured confounder. In this paper, we take a novel approach whereby the sensitivity parameter is the proportion of unmeasured confounding. We consider different assumptions on the probability of a unit being unconfounded. In each case, we derive sharp bounds on the ATE as a function of the sensitivity parameter and propose nonparametric estimators that allow flexible covariate adjustment. We construct simultaneous confidence bands around the lower and upper bounds curves similarly to Imbens and Manski (2004). We also introduce a one-number summary of a study’s robustness to the number of confounded units. Finally, we explore finite-sample properties via simulation, and apply the methods to an observational database used to assess the effects of right heart catheterization (Connors et al., 1996).