It is generally accepted that the charge transport in dielectrics is governed by coulombic trap ionization due to a barrier lowering in high electric fields (Frenkel effect). In this paper, the charge transport mechanism in Si3N4 and nonstoichiometric silicon rich SiNx is experimentally studied and quantitatively analyzed with five theoretical models: Frenkel model of Coulomb traps ionization, Hill-Adachi model of overlapping Coulomb traps, Shklovskii-Efros percolation model, Makram-Ebeid and Lannoo model of multiphonon isolated traps ionization and Nasyrov-Gritsenko model of phonon-assisted electron tunneling between nearby traps. It is shown that the charge transport in Si3N4 and SiNx is qualitatively described by Frenkel effect, but Frenkel effect predicts an enormously low attempt to escape factor value. The charge transport at traps energies W t = 1.6 eV and W opt = 3.2 eV in Si3N4 and SiNx can be described by an increase in traps concentration in the framework of Makram-Ebeid and Lannoo model and Nasyrov-Gritsenko model. The Makram-Ebeid and Lannoo model quantitatively describes the charge transport in Si3N4 and SiNx with low silicon enrichment. The charge transport in nonstoichiometric SiNx with high silicon enrichment is well explained by Nasyrov-Gritsenko model.