I argue that lottery-based school choice programs offer the opportunity to study a unique group of students: those who want to attend or are very interested in attending private school but simply cannot, even when given the chance. The differences between these students and those who choose private school are compelling education outcomes in their own right. To illustrate the argument, I analyze data from a small and little-known private school scholarship lottery in Charlotte, North Carolina, that occurred prior to the 1999-2000 academic year. I show that race, family structure, employment status, and religion significantly predict the decision to refuse a voucher offer, as does student admission into a specific school of choice. I argue that models of voucher effects on student achievement are interpretable only in the context of factors underlying the ability to choose in the first place.