In a previous Letter, we derived fundamental limits to radiative heat transfer applicable in near- through far-field regimes, based on the choice of material susceptibilities and bounding surfaces enclosing arbitrarily shaped objects; the limits exploit algebraic properties of Maxwell's equations and fundamental principles such as electromagnetic reciprocity and passivity. In this Letter, we apply these bounds to two different geometric configurations of interest, namely dipolar particles or extended structures of infinite area in the near field of one another. We find that while near-field radiative heat transfer between dipolar particles can saturate purely geometric "Landauer" limits, bounds on extended structures cannot, instead growing very slowly with respect to a material response figure of merit (an "inverse resistivity" for metals) due to the deleterious effects of multiple scattering between bodies. While nanostructuring can produce infrared resonances, it is generally unable to further enhance the resonant energy transfer spectrum beyond what is practically achieved by planar media at the surface polariton condition.