Primitive‐equation models are essential tools for studying ocean dynamics and their ever‐increasing resolution uncovers ever finer scales. At mesoscales and submesoscales, baroclinic instability is one of the main drivers of turbulence, but spurious numerical instabilities can also arise, leading to nonphysical dynamics. This study investigates a spurious instability termed Baroclinic Instability of Computational Kind (BICK), discovered in Arakawa and Moorthi (1988, https://doi.org/10.1175/1520‐0469(1988)045<1688:BIIVDS>2.0.CO;2) and Bell and White (2017, https://doi.org/10.1016/j.ocemod.2017.08.001), through idealized configurations using a vertical (Modified) Lorenz grid. Here, we explore the growth of BICK within quasi‐geostrophic (QG) and hydrostatic primitive‐equation (HPE) frameworks for different setups: the canonical Eady configuration, stratification‐modified Eady configurations, and a surface‐intensified jet configuration. Our results confirm that the emergence of BICK is specific to the vertical staggering of the (Modified) Lorenz grids. Its growth is consistent with linear QG theory, and BICK is confined near the surface and bottom boundaries. In HPE simulations, the nonlinear evolution of BICK generates small‐scale spurious eddies and reduces frontal sharpness. Increasing the number of levels reduces BICK's horizontal scale down to below the model's effective resolution. We illustrate this property using regional HPE simulations with a varying number of levels. BICK is found to significantly affect the vertically under‐resolved simulations by introducing small‐scale noise from both the bottom and surface boundaries. Our recommendation is to keep the ratio between the model horizontal and vertical resolution greater than , where is the Brunt‐Väisälä frequency and the Coriolis parameter, to minimize the impact of BICK on the dynamics.