Randomized quasi-Monte Carlo (RQMC) can produce an estimator of a mean (i.e., integral) with rootmean-square error that shrinks at a faster rate than Monte Carlo's. While RQMC is often employed to provide a confidence interval (CI) for the mean, this approach implicitly assumes that the RQMC estimator obeys a central limit theorem (CLT), which has not been established for most RQMC settings. To address this, we provide various conditions that ensure an RQMC CLT, as well as an asymptotically valid CI, and examine the tradeoffs in our restrictions. Our sufficient conditions, depending on the regularity of the integrand, generally require that the number of randomizations grows sufficiently fast relative to the number of points used from the low-discrepancy sequence.