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A full characterization of the electrical contact between conductive atomic force microscope (AFM) cantilevers and carbon nanotubes (CNTs) is presented. The dependence of current through the contact on loading force, geometric parameters, bias conditions, and time is studied in a two-terminal configuration, where a gold coated AFM cantilever serves as a movable electrode. We find that for an optimized placement of the cantilever relative to the CNT, the current through the contact becomes independent of the loading force beyond a certain limit, and this behavior is also independent of bias conditions. In that load-independent regime, the contact is stable in time to within the current fluctuations imposed by 1/f and telegraph noise in the CNT channel. Under certain conditions of tip placement, the current through the contact exhibits a non-monotonic behavior with loading, which is well explained by the parasitic planar motion of the cantilever.