We explore cosmological constraints on the sum of the three active neutrino masses Mν in the context of dynamical dark energy (DDE) models with equation of state (EoS) parametrized as a function of redshift z by w(z) = w0 + wa z/(1 + z), and satisfying w(z) ≥ −1 for all z. We make use of Cosmic Microwave Background data from the Planck satellite, Baryon Acoustic Oscillations measurements, and Supernovae Ia luminosity distance measurements, and perform a Bayesian analysis. We show that, within these models, the bounds on Mν do not degrade with respect to those obtained in the ΛCDM case; in fact the bounds are slightly tighter, despite the enlarged parameter space. We explain our results based on the observation that, for fixed choices of w0 , wa such that w(z) ≥ −1 (but not w = −1 for all z), the upper limit on Mν is tighter than the ΛCDM limit because of the well-known degeneracy between w and Mν. The Bayesian analysis we have carried out then integrates over the possible values of w0-wa such that w(z) ≥ −1, all of which correspond to tighter limits on Mν than the ΛCDM limit. We find a 95% credible interval (C.I.) upper bound of Mν < 0.13 eV. This bound can be compared with the 95% C.I. upper bounds of Mν < 0.16 eV, obtained within the ΛCDM model, and Mν < 0.41 eV, obtained in a DDE model with arbitrary EoS (which allows values of w < −1). Contrary to the results derived for DDE models with arbitrary EoS, we find that a dark energy component with w(z) ≥ −1 is unable to alleviate the tension between high-redshift observables and direct measurements of the Hubble constant H0. Finally, in light of the results of this analysis, we also discuss the implications for DDE models of a possible determination of the neutrino mass ordering by laboratory searches.