The problem of mating two parts with low clearance remains difficult for autonomous robots. We present bi-level belief assembly (BILBA), a model-based planner that computes a sequence of compliant motions which can leverage contact with the environment to reduce uncertainty and perform challenging assembly tasks with low clearance. Our approach is based on first deriving candidate contact schedules from the structure of the configuration space obstacle of the parts and then finding compliant motions that achieve the desired contacts. We demonstrate that BILBA can efficiently compute robust plans on multiple simulated tasks as well as a real robot rectangular peg-in-hole insertion task.