In this paper we use data from students who participated in the oversubscribed Boston Public Schools (BPS) prekindergarten program as a window into variation in the program's medium-term effects. We first examine whether, for the sample of students who applied to oversubscribed BPS prekindergarten programs, there is variation in the effects of the Boston prekindergarten program on children's kindergarten-through-second-grade retention, kindergarten-through-third-grade special education placement, and third-grade state test scores. We find statistically significant variation in effects on student outcomes, and we predict this variation with multiple proxies for early elementary school quality. We find that the academic proficiency of third-graders within the schools for which prekindergarten children competed is most strongly associated with prekindergarten program effects. Students who won a lottery for a prekindergarten program in a school with third-grade academic proficiency scores in the bottom quartile of the distribution experience no or negative effects by third grade. In contrast, students who won a lottery for a prekindergarten program in a school with third-grade academic proficiency scores in the top quartile of the distribution experience positive effects by third grade. An exploration of how this quality measure is defined suggests that while a part of its predictive power may be related to the characteristics of the students who enroll in these schools (specifically, their family income level), it also appears that the schools themselves contribute to these effects. Prekindergarten gains persisted if kids applied to and won a seat in a higher-quality elementary school. (author abstract)