Assessments represent an important component of undergraduate courses because they affect how students interact with course content and gauge student achievement of course objectives. To make decisions on assessment design, instructors must understand the affordances and limitations of available question formats. Here, we use a crossover experimental design to identify differences in how multiple-true-false (MTF) and free-response (FR) exam questions reveal student thinking regarding specific conceptions. We report that correct response rates correlate across the two formats but that a higher percentage of students provide correct responses for MTF questions. We find that MTF questions reveal a high prevalence of students with mixed (correct and incorrect) conceptions, while FR questions reveal a high prevalence of students with partial (correct and unclear) conceptions. These results suggest that MTF question prompts can direct students to address specific conceptions but obscure nuances in student thinking and may overestimate the frequency of particular conceptions. Conversely, FR questions provide a more authentic portrait of student thinking but may face limitations in their ability to diagnose specific, particularly incorrect, conceptions. We further discuss an intrinsic tension between question structure and diagnostic capacity and how instructors might use multiple formats or hybrid formats to overcome these obstacles.