By using a rather simple model potential we calculate the influence of specular and diffuse scattering at interfaces on the resistance for currents perpendicular to the plane of the layers (CPP) in multilayered metallic structures. We find that if one maintains the same interfaces between layers, their contribution to the total resistance of the system depends on the thickness of the intervening layers. Their contribution is constant only if the distance between interface is larger than the electron mean free path in the bulk of the layers. The error incurred by neglecting the dependence of the interface resistance on layer thickness depends on the amount of diffuse scattering at interfaces and the height of the potential step at the interface. For a Fe-Cr or Co-Cu multilayer consisting of three or five layers with realistic interfaces where both specular and diffuse scattering may occur, the error reaches 15-20%; this exceeds the experimental error. We suggest systems where this effect may be observed.