Large Language Models (LLMs) can generate code, but can they generate fast code for complex, real-world software systems? In this study, we investigate this question using a dataset of 65 tasks mined from performance-critical open-source Java projects. Unlike prior studies, which focused on algorithmic puzzles, we conduct experiments on actual performance-sensitive production code and employ developer-written JMH benchmarks to rigorously validate performance gains against human baselines. Our results reveal a nuanced reality -- although LLMs demonstrate a surprisingly high capability to solve these complex engineering problems, their solutions suffer from extreme volatility and still lag behind human developers on average. Consequently, we find that the current benchmarks based on algorithmic tasks yields an overly optimistic assessment of LLM capabilities. We trace this real-world performance gap to two primary limitations: first, LLMs struggle to autonomously pinpoint performance hotspots, and second, even with explicit guidance, they often fall short of synthesizing optimal algorithmic improvements. Our results highlight the need to move beyond static code generation towards more complex agent-based systems that are able to profile and observe runtime behavior for performance improvement.