Puns are a form of humorous wordplay that exploits polysemy and phonetic similarity. While LLMs have shown promise in detecting puns, we show in this paper that their understanding often remains shallow, lacking the nuanced grasp typical of human interpretation. By systematically analyzing and reformulating existing pun benchmarks, we demonstrate how subtle changes in puns are sufficient to mislead LLMs. Our contributions include comprehensive and nuanced pun detection benchmarks, human evaluation across recent LLMs, and an analysis of the robustness challenges these models face in processing puns.