We present a methodology to systematically test conversational recommender systems with regards to conversational breakdowns. It involves examining conversations generated between the system and simulated users for a set of pre-defined breakdown types, extracting responsible conversational paths, and characterizing them in terms of the underlying dialogue intents. User simulation offers the advantages of simplicity, cost-effectiveness, and time efficiency for obtaining conversations where potential breakdowns can be identified. The proposed methodology can be used as diagnostic tool as well as a development tool to improve conversational recommendation systems. We apply our methodology in a case study with an existing conversational recommender system and user simulator, demonstrating that with just a few iterations, we can make the system more robust to conversational breakdowns.