Models of dark energy in which neutrinos interact with the scalar field supposed to be responsible for the acceleration of the Universe usually imply a variation of the neutrino masses on cosmological time scales. In this work we propose a parametrization for the neutrino mass variation that captures the essentials of those scenarios and allows one to constrain them in a model independent way, that is, without resorting to any particular scalar field model. Using WMAP 5 yr data combined with the matter power spectrum of SDSS and 2dFGRS, the limit on the present value of the neutrino mass is m(0) equivalent to m(nu)(z = 0) < 0.43 (0.28) eV at 95% C.L. for the case in which the neutrino mass was lighter ( heavier) in the past, a result competitive with the ones imposed for standard (i.e., constant mass) neutrinos. Moreover, for the ratio of the mass variation of the neutrino mass Delta m(nu) over the current mass m(0) we found that log[vertical bar Delta m(nu)vertical bar/m(0)] < -1.3 (-2.7) at 95% C.L. for Delta m(nu) < 0 (Delta m(nu) > 0), totally consistent with no mass variation. These stringent bounds on the mass variation are not related to the neutrino freestreaming history which may affect the matter power spectrum on small scales. On the contrary, they are imposed by the fact that any significant transfer of energy between the neutrino and dark energy components would lead to an instability contradicting CMB and large-scale structure data on the largest observable scales.