Boussinesq type (BT) models play a major role in coastal engineering applications. Due to their asymptotic nature, these models have lent themselves to constant study, and improvement based on the inclusion/removal of asymptotically small (or not) terms. In this work we go back to the fundamental question of whether full nonlinear in weakly dispersive BT models is a necessity. We reconsider the tests by [67] to address this issue, as well as a number of more demanding ones. We also consider different families of weakly nonlinear BT models, with different shoaling characteristics, especially when nonlinear waves are involved. Our study allows to point out that for many cases it is quite hard to conclude in favor of one class of model or another. There are a few discriminant cases, which are unfortunately not those mostly used in the literature proposing new models or new numerical methods.