We consider a "metal" with a mean free path possessing a sharp maximum at the Fermi surface. At finite temperature, the width of this maximum is close to kBT. We show that the electrical conductivity of this system behaves as if the peak of the mean free path remains sharp at finite temperatures. In contrast to the well-known normal situation, the resistivity due to elastic scattering is surprisingly found to be linear in temperature, and the resistivity due to scattering by phonons is proportional to T2. This model is proposed to relate to some properties of "doped insulators", such as cuprates, organic metals and fullerenes.