Conventional methods of cluster-robust inference are inconsistent in the presence of unignorably large clusters. We formalize this claim by establishing a necessary and sufficient condition for the consistency of the conventional methods. We find that this condition for the consistency is rejected for a majority of empirical research papers. In this light, we propose a novel score subsampling method that achieves uniform size control over a broad class of data generating processes, covering that fails the conventional method. Simulation studies support these claims. With real data used by an empirical paper, we showcase that the conventional methods conclude significance while our proposed method concludes insignificance.