Since the single-ion thermodynamic properties of bulk solutions are not directly accessible from experiments, extrapolations have been devised to estimate them from experimental measurements on small-clusters. Extrapolations based on the cluster-pair-based approximation (CPA) technique (Tissandier et al. J. Phys. Chem. A 1998, 102, 7787-7794) and its variants are currently considered one of the most reliable source of single-ion hydration thermodynamic data and have been used as a benchmark for the development of molecular and continuum solvation models. Despite its importance, the CPA has not been thoroughly tested and recent studies have indicated inconsistencies with molecular simulations. The present work challenges the key CPA assumptions that the hydration properties of single cations and anions in growing clusters rapidly converge to each other following a monotonous trend. Using a combination of simulation techniques to study the transition between alkali halide ions in small clusters and bulk solution, we show that this convergence is rather slow and involves a surprising change in trends, which can result in significant errors in the original estimated single-ion properties. When these cluster-size-dependent effects are taken into account, the inconsistencies between molecular models and experimental predictions disappear, and the value of the proton hydration enthalpy based on the CPA aligns with estimates based on other principles.