We numerically analyze the process of dip coating on fibers when the liquid to be deposited contains soluble surfactants. The predictions obtained are in excellent agreement with experimental results reported by different authors. Our results show that the thickness of the film deposited, and particularly the so-called surfactant thickening effect, changes its behavior when the coating speed exceeds certain limits; therefore, the effects produced by inertia forces are scrutinized to uncover, for the first time, the mechanisms inducing those changes.