The non-ohmic behavior of Nafion electrical properties, i. e., the thickness and potential dependent conductivity, was studied in the impedance, dielectric and conductivity representations with the use of a special through-plane sampleholder in a 4-probe array. Such measurements allowed identifying the genuine ac-to-dc conductivity crossover frequency in Nafion, which occurs for f<10-1 Hz. In addition, the minimization of the interfacial electrode/ionomer polarizations with the 4-probe setup permitted the determination of the bulk dc conductivity and dielectric constant of Nafion, which are ??~0.03 Scm-1 and ?????~106 (T=40 ??Cand RH=100%), respectively. The colossal dielectric constant is shown to increase the Debye length of the electric double layer to values comparable to the membrane thickness. Therefore, the exponential increase of the proton conductivity with increasing both membrane thickness and electric potential are a result of canceling out the non-linear effects of electric double layer caused by the high dielectric permittivity of Nafion. The ac-to-dc conductivity crossover in H2/O2 fuel cell impedance curves takes place for f<100 Hz and matches with the ex situ impedance spectroscopy study in excellent agreement, revealing a striking result: the potential dependent conductivity of Nafion requires extra fuel cell overpotential to overcome the electrode/ionomer interfacial polarization representing an additional polarization loss to polymer electrolyte fuel cells.