We investigate the impact of prior models on the upper bound of the sum of neutrino masses, ∑m_{ν}. Using data from the large scale structure of galaxies, cosmic microwave background, type Ia supernovae, and big bang nucleosynthesis, we argue that cosmological neutrino mass and hierarchy determination should be pursued using exact models, since approximations might lead to incorrect and nonphysical bounds. We compare constraints from physically motivated neutrino mass models (i.e., ones respecting oscillation experiments) to those from models using standard cosmological approximations. The former give a consistent upper bound of ∑m_{ν}≲0.26  eV (95% CI) and yield the first approximation-independent upper bound for the lightest neutrino mass species, m_{0}^{ν}<0.086  eV (95% CI). By contrast, one of the approximations, which is inconsistent with the known lower bounds from oscillation experiments, yields an upper bound of ∑m_{ν}≲0.15  eV (95% CI); this differs substantially from the physically motivated upper bound.