LLM agents operating under organizational policies must comply with authorization constraints typically specified in natural language. In practice, such specifications inevitably contain ambiguities and logical or semantic gaps that cause the agent's behavior to systematically diverge from the true requirements. We ask: by letting an agent evolve its policy understanding through interaction and corrective feedback from pre-deployment testing, can it autonomously refine its interpretation to close specification gaps? We propose PolicyBank, a memory mechanism that maintains structured, tool-level policy insights and iteratively refines them -- unlike existing memory mechanisms that treat the policy as immutable ground truth, reinforcing "compliant but wrong" behaviors. We also contribute a systematic testbed by extending a popular tool-calling benchmark with controlled policy gaps that isolate alignment failures from execution failures. While existing memory mechanisms achieve near-zero success on policy-gap scenarios, PolicyBank closes up to 82% of the gap toward a human oracle.