Previous research has estimated that food insecure children are more likely to suffer from a wide array of negative health outcomes than food secure children, leading many to claim that alleviating food insecurity would lead to better health outcomes. Identifying the causal impacts is problematic, however, given endogenous selection into food security status and potential mismeasurement of true food security status. Using recently developed nonparametric bounding methods and data from the 2001-2006 National Health and Nutritional Examination Survey (NHANES), we assess what can be identified about the effects of food insecurity on child health outcomes in the presence of nonrandom selection and nonclassical measurement error. Under relatively weak monotonicity assumptions, we can identify that food security has a statistically significant positive impact on good general health and being a healthy weight. Our work suggests that previous research has more likely underestimated than overestimated the causal impacts of food insecurity on health.