Measurements are presented from a two-beam structure with several bolted interfaces in order to characterize the nonlinear damping introduced by the joints. The measurements (all at force levels below macroslip) reveal that each underlying mode of the structure is well approximated by a single degree-of-freedom (SDOF) system with a nonlinear mechanical joint. At low enough force levels, the measurements show dissipation that scales as the second power of the applied force, agreeing with theory for a linear viscously damped system. This is attributed to linear viscous behavior of the material and/or damping provided by the support structure. At larger force levels, the damping is observed to behave nonlinearly, suggesting that damping from the mechanical joints is dominant. A model is presented that captures these effects, consisting of a spring and viscous damping element in parallel with a four-parameter Iwan model. The parameters of this model are identified for each mode of the structure and comparisons suggest that the model captures the stiffness and damping accurately over a range of forcing levels.