Deficiencies in the parameterizations of convection used in global climate models often lead to a distorted representation of the simulated rainfall intensity distribution (i.e., too much rainfall from weak rain rates). While encouraging improvements in high percentile rainfall intensity have been found as the horizontal resolution of the Community Atmosphere Model is increased to ∼25 km, we demonstrate no corresponding improvement in the moderate rain rates that generate the majority of accumulated rainfall. Using a statistical framework designed to emphasize links between precipitation intensity and accumulated rainfall beyond just the frequency distribution, we show that CAM cannot realistically simulate moderate rain rates, and cannot capture their intensification with climate change, even as resolution is increased. However, by separating the parameterized convective and large‐scale resolved contributions to total rainfall, we find that the intensity, geographic pattern, and climate change response of CAM's large‐scale rain rates are more consistent with observations (TRMM 3B42), superparameterization, and theoretical expectations, despite issues with parameterized convection. Increasing CAM's horizontal resolution does improve the representation of total rainfall intensity, but not due to changes in the intensity of large‐scale rain rates, which are surprisingly insensitive to horizontal resolution. Rather, improvements occur through an increase in the relative contribution of the large‐scale component to the total amount of accumulated rainfall. Analysis of sensitivities to convective timescale and entrainment rate confirm the importance of these parameters in the possible development of scale‐aware parameterizations, but also reveal unrecognized trade‐offs from the entanglement of precipitation frequency and total amount.