While the baryon asymmetry of the Universe is nowadays well measured by cosmological observations, the bounds on the lepton asymmetry in the form of neutrinos are still significantly weaker. We place limits on the relic neutrino asymmetries using some of the latest cosmological data, taking into account the effect of flavor oscillations. We present our results for two different values of the neutrino mixing angle theta(13), and show that for large theta(13) the limits on the total neutrino asymmetry become more stringent, diluting even large initial flavor asymmetries. In particular, we find that the present bounds are still dominated by the limits coming from big bang nucleosynthesis, while the limits on the total neutrino mass from cosmological data are essentially independent of theta(13). Finally, we perform a forecast for Cosmic Origins Explorer, taken as an example of a future cosmic microwave background experiment, and find that it could improve the limits on the total lepton asymmetry approximately by up to a factor 6.6.