We report interferometric measurements of [N II] 205 μm fine-structure line emission from a representative sample of three galaxies at z = 5– 6 using the Atacama Large (sub)Millimeter Array (ALMA). These galaxies were previously detected in [C II] and far-infrared continuum emission and span almost two orders of magnitude in star formation rate (SFR). Our results show at least two different regimes of ionized interstellar medium properties for galaxies in the first billion years of cosmic time, separated by their L_([C II])/L_([N II]) ratio. We find extremely low [N II] emission compared to [C II] (L_([C II])/L_([N II]) = 68^(+200)_(-28)) from a "typical" ~L_*(UV) star-forming galaxy, likely directly or indirectly (by its effect on the radiation field) related to low dust abundance and low metallicity. The infrared-luminous modestly star-forming Lyman-break galaxy (LBG) in our sample is characterized by an ionized-gas fraction (L_([C II])/L_([N II]) ≾ 20) typical of local star-forming galaxies and shows evidence for spatial variations in its ionized-gas fraction across an extended gas reservoir. The extreme SFR, warm and compact dusty starburst AzTEC-3 shows an ionized fraction higher than expected given its SFR surface density (L_([C II])/L_([N II]) = 22 ± 8) suggesting that [N II] dominantly traces a diffuse ionized medium rather than star-forming H II regions in this type of galaxy. This highest redshift sample of [N II] detections provides some of the first constraints on ionized and neutral gas modeling attempts and on the structure of the interstellar medium at z = 5–6 in "normal" galaxies and starbursts.