We develop and estimate a dynamic model of household labor supply and SNAP participation that explains two empirical puzzles: incomplete program take-up despite substantial benefits, and minimal employment effects of SNAP work requirements despite the constraints they impose. Our model incorporates detailed program rules, labor market frictions, and heterogeneous participation costs, revealing that eligible households may rationally forgo benefits when facing administrative barriers to participation and uncertain future employment. Exploiting our model estimates, we perform policy experiments finding that more stringent work requirements lead to dramatic reductions in participation (from 10% to 2.3%) without increasing employment levels or participants' self-sufficiency. While more stringent work requirements do not seem to be effective in our population, other policies do. An increase in benefits would lead to better labor market outcomes, while a lower benefit reduction rate would be one of the most effective policies to increase take-up.