Estimating the size of a hidden finite set is an important task in a wide variety of scientific fields. Often it is infeasible to directly enumerate the elements, and statisticians have developed various indirect approaches using structural assumptions about ordering or network relationships between units. However, evaluating the large-sample and large-population properties of these estimators can be challenging. Most popular estimators are consistent under an "infill" asymptotic regime in which the population size is fixed, while the number of samples grows. But this scenario may be starkly different from the constraints in practice. We formalize two asymptotic regimes, "infill" and "outfill", to study the large-sample behaviors of those methods. The outfill regime allows the population size to grow together with the sample size, with the latter approaching a fixed proportion of the former. This framework provides a unifying perspective for understanding asymptotic claims on hidden set size estimators. We derive the asymptotic properties of popular estimators and show that, under the outfill setting, inconsistency is a common problem from which almost all estimators suffer.