The literature assessing the efficacy of the Supplemental Nutrition Assistance Program (SNAP), formerly known as the Food Stamp Program, has long puzzled over positive associations between SNAP receipt and various undesirable health outcomes such as food insecurity. Assessing the causal impacts of SNAP, however, is hampered by two key identification problems: endogenous selection into participation and extensive systematic underreporting of participation status. Using data from the National Health and Nutrition Examination Survey (NHANES), we extend partial identification bounding methods to account for these two identification problems in a single unifying framework. Specifically, we derive informative bounds on the average treatment effect of SNAP on child food insecurity, general poor health, obesity, and anemia across a range of different assumptions used to address the selection and classification error problems. In particular, to address the selection problem we apply relatively weak nonparametric assumptions on the latent outcomes, selected treatments, and observed covariates. To address the classification error problem, we formalize a new approach that uses auxiliary administrative data on the size of the SNAP caseload to restrict the magnitudes and patterns of SNAP reporting errors. Layering successively stronger assumptions, an objective of our analysis is to make transparent how the strength of the conclusions varies with the strength of the identifying assumptions. Under the weakest restrictions, there is substantial ambiguity: we cannot rule out the possibility that SNAP increases or decreases poor health. Under stronger but plausible assumptions used to address the selection and classification error problems, we find that commonly cited relationships between SNAP and poor health outcomes provide a misleading picture about the true impacts of the program. Our tightest bounds identify favorable impacts of SNAP on child health.