We study the possible effects of large Lorentz violations that can appear in the effective models in which the Lorentz symmetry breakdown is performed with higher-order operators. For this we consider the Myers and Pospelov extension of QED with dimension-five operators in the photon sector and standard fermions. We focus on the fermion self-energy at one-loop order and find small and finite radiative corrections in the even CPT sector. In the odd CPT sector a lower dimensional operator is generated which contains unsuppressed effects of Lorentz violation leading to a possible fine-tuning. For the calculation of divergent diagrams we use dimensional regularization and consider an arbitrary background four-vector.