It is commonly assumed that shallow flows are in good approximation two-dimensional (2D) or quasi-2D. We will provide evidence that this is not always the case, and that the simple scaling argument based on the continuity equation does not always apply. Laboratory experiments on vortex flows in shallow fluid layers have revealed that locally significant three-dimensional (3D) effects and substantial vertical motions may occur, clearly destroying the assumed 2D character of the flow. For example, in the case of a dipolar vortex structure, an oscillatory vertical motion is observed in the vortex cores, while a spanwise circulation roll is present in front of the travelling dipole. These laboratory observations are confirmed by 3D numerical flow simulations. Attention will be given to a correct scaling analysis, in which both the aspect ratio of the fluid depth and a typical horizontal scale and the Reynolds number play a role.