The simplest unified extension of the minimal supersymmetric standard model with bilinear R-parity violation naturally predicts a hierarchical neutrino mass spectrum, suitable to explain atmospheric and solar neutrino fluxes. We study whether the individual violation of the lepton numbers L-e,L-mu,L-tau in the charged sector can lead to measurable rates for BR(mu --> egamma) and BR(tau --> mugamma). We find that some of the R-parity violating terms that are compatible with the observed atmospheric neutrino oscillations could lead to rates for mu --> egamma measurable in projected experiments. However, the Deltam(12)(2) obtained for those parameters is too high to be compatible with the solar neutrino data, excluding therefore the possibility of having measurable rates for mu --> egamma in the model.