Abstract: We propose here a novel method which singles out the a priori unavoidable dependence on the underlying cosmological model when extracting parameter constraints, providing robust limits which only depend on the considered dataset. Interestingly, when dealing with several possible cosmologies and interpreting the Bayesian preference in terms of the Gaussian statistical evidence, the preferred model is much less favored than when only two cases are compared. As a working example, we apply our approach to the cosmological neutrino mass bounds, which play a fundamental role not only in establishing the contribution of relic neutrinos to the dark matter of the Universe but also in the planning of future experimental searches of the neutrino character and of the neutrino mass ordering.