Nebular-phase observations and spectral models of Type Ic superluminous supernovae (SLSNe) are presented. LSQ14an and SN 2015bn both display late-time spectra similar to galaxy-subtracted spectra of SN 2007bi, and the class shows strong similarity with broad-lined SNe Ic such as SN 1998bw. Near-infrared observations of SN 2015bn show a strong Ca II triplet, O I 9263, O I 1.13 mu m, and Mg I 1.50 mu m, but no distinct He, Si, or S emission. The high Ca II NIR/[Ca II] 7291, 7323 ratio of similar to 2 indicates a high electron density of n(e) greater than or similar to 10(8) cm(-3). Spectral models of oxygen-zone emission are investigated to put constraints on the emitting region. Models require M(O-zone) greater than or similar to 10 M-circle dot to produce enough [O I] 6300, 6364 luminosity, irrespective of the powering situation and the density. The high oxygen-zone mass, supported by high estimated magnesium masses, points to explosions of massive CO cores, requiring M-ZAMS greater than or similar to 40M(circle dot). Collisions of pair-instability pulsations do not provide enough mass to account for the emission. [O II] and [O III] lines emerge naturally in many models, which strengthens the identification of broad [O II] 7320, 7330, [O III] 4363, and [O III] 4959, 5007 in some spectra. A small filling factor f less than or similar to 0.01 for the O/Mg zone is needed to produce enough luminosity in Mg I] 4571, Mg I 1.504 mu m, and O I recombination lines, which shows that the ejecta is clumped. We review the constraints from the nebular spectral modeling in the context of the various scenarios proposed for SLSNe.