USDA operates several food assistance programs aimed at alleviating food insecurity. Little is known about how they interact. We focus on SNAP and WIC, two of the largest means-tested programs that provide resources to low-income households to purchase food and differ in several respects. Our question is the extent to which participation in both programs alleviates food insecurity compared with participation in SNAP alone. We bound underlying causal effects by applying nonparametric treatment effect methods that allow for endogenous selection and underreported program participation to data from the National Household Food Acquisition and Purchase Survey (FoodAPS). FoodAPS contains administrative data to validate SNAP participation and data on the local food environment, including the cost of food, allowing us to tighten bounds on the causal effects. Under relatively weak assumptions about the selection process, combined with a food expenditure-based monotone instrumental variable, we identify that the marginal impact of participating in both programs is strictly positive. This finding provides evidence that the programs are nonredundant, which can aid policymakers in improving the design and targeting of food assistance programs. The methods showcase what can be learned about treatment effects when validation data are available for one program but not the other.