Abstract: The most recent measurements of the temperature and low-multipole polarization anisotropies of the cosmic microwave background from the Planck satellite, when combined with galaxy clustering data from the Baryon Oscillation Spectroscopic Survey in the form of the full shape of the power spectrum, and with baryon acoustic oscillation measurements, provide a 95% confidence level (C.L.) upper bound on the sum of the three active neutrinos Sigma m(nu) < 0.183 eV, among the tightest neutrino mass bounds in the literature, to date, when the same data sets are taken into account. This very same data combination is able to set, at similar to 70% C.L., an upper limit on Sigma m(nu) of 0.0968 eV, a value that approximately corresponds to the minimal mass expected in the inverted neutrino mass hierarchy scenario. If high-multipole polarization data from Planck is also considered, the 95% C.L. upper bound is tightened to Sigma m(nu) < 0.176 eV. Further improvements are obtained by considering recent measurements of the Hubble parameter. These limits are obtained assuming a specific nondegenerate neutrino mass spectrum; they slightly worsen when considering other degenerate neutrino mass schemes. Low-redshift quantities, such as the Hubble constant or the reionization optical depth, play a very important role when setting the neutrino mass constraints. We also comment on the eventual shifts in the cosmological bounds on Sigma m(nu) when possible variations in the former two quantities are addressed.