A single-molecule break junction device serves as a tunable model system for probing the many-body Kondo state. The low-energy properties of this state are commonly described in terms of a Kondo model, where the response of the system to different perturbations is characterized by a single emergent energy scale, kBTK. Comparisons between different experimental systems have shown issues with numerical consistency. With a new constrained analysis examining the dependence of conductance on temperature, bias, and magnetic field simultaneously, we show that these deviations can be resolved by properly accounting for background, non-Kondo contributions to the conductance that are often neglected. We clearly demonstrate the importance of these non-Kondo conduction channels by examining transport in devices with total conductances exceeding the theoretical maximum due to Kondo-assisted tunneling alone. © 2013 American Physical Society.