We perform a new study of future sensitivities of galaxy redshift surveys to the free-streaming effect caused by neutrino masses, adding the information on cosmological parameters from measurements of primary anisotropies of the cosmic microwave background (CMB). Our reference cosmological scenario has nine parameters and three different neutrino masses, with a hierarchy imposed by oscillation experiments. Within the present decade, the combination of the Sloan Digital Sky Survey (SDSS) and CMB data from the PLANCK experiment will have a 2-sigma detection threshold on the total neutrino mass close to 0.2 eV. This estimate is robust against the inclusion of extra free parameters in the reference cosmological model. On a longer term, the next generation of experiments may reach values for the total neutrino mass of order 0.1 eV at 2-sigma, or better if a galaxy redshift survey significantly larger than SDSS is completed. We also discuss how the small changes on the free-streaming scales in the normal and inverted hierarchy schemes are translated into the expected errors from future cosmological data.