This paper raises the question of whether the Kramers-Kronig relationships place severe limitations on the bandwidths that can be achieved in homogeneous microwave absorbing materials. Secondly, the paper attempts to determine, by computer simulation, the best bandwidths theoretically achievable for homogeneous coatings of one particular class of materials: namely, materials whose magnetic properties are defined by the Gilbert equation. Lastly, the paper reports on the beginnings of an effort to use computer simulation to determine the maximum bandwidths achievable for multilayer absorbing structures whose constitutive parameters not only satisfy the requirements of Kramers-Kronig, but also are ''reasonable'' from an engineering standpoint.