Large language model (LLM) agents increasingly coordinate in multi-agent systems, yet we lack an understanding of where and why cooperation fails. Many real-world coordination problems are not social dilemmas: helping others -- sharing documentation, unblocking a teammate -- costs the helper almost nothing while producing substantial collective benefit. Whether LLM agents cooperate in this regime, where helping is free and they are explicitly instructed to do so, remains unknown. We build a turn-based multi-agent environment that strips away all strategic complexity, making cooperation costless and trivially optimal. Across eight widely used LLMs, capability does not predict cooperation: OpenAI o3 reaches only 17% of optimal collective performance while the weaker o3-mini reaches 50%, despite identical instructions to maximize group revenue. Using a causal decomposition that automates one side of agent communication, we separate cooperation failures from competence failures, and find that several capable models actively withhold information despite gaining nothing from withholding. Targeted interventions address each mode: explicit protocols roughly double the performance of competence-limited models, while small sharing incentives unlock cooperation-limited ones. Our results suggest that scaling intelligence alone will not solve coordination in multi-agent systems, and will require deliberate cooperative design, even when helping costs nothing.