Discretized numerical models of the atmosphere are usually intended to faithfully represent an underlying set of continuous equations, but sometimes fails to do so because of subtle pathologies that have crept into the discretized equations. In this study, convergence testing is employed to verify the discretization of the Cloud Layers Unified By Binormals (CLUBB) single-column model (SCM) of clouds and turbulence. Convergence testing reveals two types of pathologies in CLUBB-SCM. First, the non-linear diffusion term added to stabilize the simulation amplifies rather than reduces the non-smoothness in the model solution in some cases. Second, numerical limiters (i.e., clipping) used by CLUBB introduce discontinuities or slope discontinuities in model fields, leading to additional non-smoothness in model solution. These pathologies cause the numerical solution from CLUBB to not converge at the expected rate with increasingly refined spatial and temporal resolutions. As a remedy, we found that smoothing the limiters and using linear diffusion can reduce the non-smoothness and restore the expected first-order convergence in CLUBB’s solutions for some test cases. These model reformulations also improve the results at coarser, near-operational grid spacing and time steps in cumulus cloud and dry turbulence tests. The application in our study demonstrates that convergence testing is a valuable tool for detecting pathologies, including unintended discontinuities and grid dependence, in the model equation set.