The current density-voltage (J-V) characteristics of hole-only diodes based on poly(2-methoxy, 5-(2' ethylhexyloxy)-p-phenylene vinylene) (MEH-PPV) were measured at a wide temperature and field range. At high electric fields the temperature dependence of the transport vanishes, and all J-V sweeps converge to a power law. Nuclear tunneling theory predicts a power law at high fields that scales with the Kondo parameter. To model the J -V characteristics we have performed master-equation calculations to determine the dependence of charge carrier mobility on electric field, charge carrier density, temperature, and Kondo parameter, using nuclear tunneling transfer rates. We demonstrate that nuclear tunneling, unlike other semiclassical models, p...