The neutrino mixing matrix U is in general of the form U=U_e^+ U_nu, where U_e arises from the diagonalization of charged leptons and U_nu is from the neutrino sector. We discuss the possibility that U_nu is nearly diagonal (in the lagrangian basis) and the observed mixing arises with good accuracy from U_e. We find that the fact that, in addition to the nearly maximal atmospheric mixing angle theta_{23}, the solar angle theta_{12} is definitely also large while at the same time the third mixing angle theta_{13} is small, makes the construction of a natural model of this sort considerably more complicated. We present an example of a natural model of this class. We also find that the case that U_nu is exactly of the bimixing type is severely constrained by the bound on theta_{13} but not excluded. We show that planned experimental searches for theta_{13} could have a strong impact on bimixing models.