Mixtures of room temperature ionic liquids (IL) with neutral organic molecules provide a valuable testing ground to investigate the interplay of the ionic and molecular-dipolar state in dense Coulomb systems at near ambient conditions. In the present study, the viscosity eta and the ionic conductivity a of 1-n-butyl-3-methylimidazolium hexafluorophosphate ([bmim][PF6])/naphthalene mixtures at T = 80 degrees C have been measured at 10 stoichiometries spanning the composition range from pure naphthalene to pure [bmim][PF6]. The viscosity grows nearly monotonically with increasing IL mole fraction (x), whereas the conductivity per ion displays a clear peak at x approximate to 15%. The origin of this maximum has been investigated using molecular dynamics simulations based on a classical force field. Snapshots of the simulated samples show that the conductivity maximum is due to the gradual transition in the IL component from an ionic state at high x to a dipolar fluid made of neutral ion pairs at low x. At concentrations x < 0.20 the ion pairs condense into molecular-thin filaments bound by dipolar forces and extending in between nanometric droplets of IL. These results are confirmed and complemented by the computation of dynamic and transport properties in [bmim][PF6]/naphthalene mixtures at low IL concentration.