Tunnel junctions often have conductances that are linear in voltage over wide voltage ranges. Although this linear conductance background has received much attention recently in connection with the high-Tc superconductors, it is found in many systems. Previous work has shown that the discontinuity in slope of the linear conductance background at zero bias is thermally smeared by 5kT, consistent with inelastic tunneling from a broad, flat continuum of states. In this paper we concentrate on a particular system, junctions with Cr2O3 in the barrier region, in which the size of the effect can be adjusted by changing the barrier composition. Careful analysis of the Pb strong-coupling phonon structure in Al-Cr2O3-Pb junctions is consistent with the inelastic-tunneling hypothesis, and inconsistent with a voltage-dependent matrix-element explanation for this effect. Suppression of the Al gap signature, as well as Zeeman splitting of the conductance background near zero bias, indicates that strong spin interactions occur in the barrier region in these junctions. We present Monte Carlo simulations of multiple-scattering strong inelastic tunneling to explain the very wide voltage range over which the linear background occurs. We argue that the same basic mechanism is probably responsible for the linear background in junctions involving high-Tc junctions as for the junctions studied in detail here. © 1992 The American Physical Society.