We investigate the validity and accuracy of two commonly-used two-dimensional models of a rotating, incompressible, shallow fluid layer: the shallow-water (SW) model and the Serre/Green-Naghdi (GN) model. The models differ only in one respect: the SW model imposes the hydrostatic approximation while the GN model does not. By comparing with the parent three-dimensional (3D) model (Euler's equations with a free surface), we find that GN is significantly more accurate except at small scales, where the commonly-used form of the equations are ill-suited computationally. A new explicit form of the equations, using the vertically-integrated non-hydrostatic pressure, overcomes this problem and proves remarkably robust and accurate. Implications for modelling stratified flows are discussed.