Can neutrino mixings arise from the charged lepton sector?

The neutrino mixing matrix U is in general of the form U=Ue†Uν, where Ue arises from the diagonalization of charged leptons and Uν is from the neutrino sector. We discuss the possibility that Uν is nearly diagonal (in the Lagrangian basis) and the observed mixing arises with good accuracy from Ue. We find that the fact that, in addition to the nearly maximal atmospheric mixing angle θ23, the solar angle θ12 is definitely also large while at the same time the third mixing angle θ13 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 Uν is exactly of the bimixing type is severely constrained by the bound on θ13 but not excluded. We show that planned experimental searches for θ13 could have a strong impact on bimixing models