Although concept inventories are among the most frequently used tools in the physics and astronomy education communities, they are rarely evaluated using item response theory (IRT). When IRT models fit the data, they offer sample-independent estimates of item and person parameters. IRT may also provide a way to measure students' learning gains that circumvents some known issues with Hake's normalized gain. In this paper, we review the essentials of IRT while simultaneously applying it to the Star Properties Concept Inventory. We also use IRT to explore an important psychometrics debate that has received too little attention from physics and astronomy education researchers: What do we mean when we say we "measure" a mental process? This question leads us to use IRT to address the provocative question that constitutes the title of this paper: Do concept inventories actually measure anything? (Contains 18 figures and 9 tables.)