  We study the interplay of conductive and radiative heat transfer (RHT) in
planar geometries and predict that temperature gradients induced by radiation
can play a significant role on the behavior of RHT with respect to gap sizes,
depending largely on geometric and material parameters and not so crucially on
operating temperatures. Our findings exploit rigorous calculations based on a
closed-form expression for the heat flux between two plates separated by vacuum
gaps $d$ and subject to arbitrary temperature profiles, along with an
approximate but accurate analytical treatment of coupled conduction--radiation
in this geometry. We find that these effects can be prominent in typical
materials (e.g. silica and sapphire) at separations of tens of nanometers, and
can play an even larger role in metal oxides, which exhibit moderate
conductivities and enhanced radiative properties. Broadly speaking, these
predictions suggest that the impact of RHT on thermal conduction, and vice
versa, could manifest itself as a limit on the possible magnitude of RHT at the
nanoscale, which asymptotes to a constant (the conductive transfer rate when
the gap is closed) instead of diverging at short separations.
