  We study the decay rates of the $\mu \to e \gamma$ and $\tau \to \ell \gamma$
transitions in the framework of the type III seesaw model, where fermionic
triplets are exchanged to generate neutrino masses. We show that the
observation of one of those decays in planned experiments would contradict
bounds arising from present experimental limits on the $\mu \to eee$ and $\tau
\to 3 l$ decay rates, and therefore imply that there exist other sources of
lepton flavour violation than those associated to triplet of fermions.
