Failures in large-scale cloud systems incur substantial financial losses, making automated Root Cause Analysis (RCA) essential for operational stability. Recent efforts leverage Large Language Model (LLM) agents to automate this task, yet existing systems exhibit low detection accuracy even with capable models, and current evaluation frameworks assess only final answer correctness without revealing why the agent's reasoning failed. This paper presents a process level failure analysis of LLM-based RCA agents. We execute the full OpenRCA benchmark across five LLM models, producing 1,675 agent runs, and classify observed failures into 12 pitfall types across intra-agent reasoning, inter-agent communication, and agent-environment interaction. Our analysis reveals that the most prevalent pitfalls, notably hallucinated data interpretation and incomplete exploration, persist across all models regardless of capability tier, indicating that these failures originate from the shared agent architecture rather than from individual model limitations. Controlled mitigation experiments further show that prompt engineering alone cannot resolve the dominant pitfalls, whereas enriching the inter-agent communication protocol reduces communication-related failures by up to 15 percentage points. The pitfall taxonomy and diagnostic methodology developed in this work provide a foundation for designing more reliable autonomous agents for cloud RCA.