Policymakers have been puzzled to observe that food stamp households appear more likely to be food insecure than observationally similar eligible nonparticipating households. We reexamine this issue allowing for nonclassical reporting errors in food stamp participation and food insecurity. Extending the literature on partially identified parameters, we introduce a nonparametric framework that makes transparent what can be known about conditional probabilities when a binary outcome and conditioning variable are both subject to nonclassical measurement error. We find that the food insecurity paradox hinges on assumptions about the data that are not supported by the previous food stamp participation literature. (Contains 5 tables, 4 figures and 22 footnotes.)