The regularities in the behavior of the electrophysical and some thermophysical properties of composite ceramic materials containing both micro- and nanoparticles of nickel in the region of the percolation threshold have been studied. Certain regularities in the behavior of the electrical conductivity, dielectric constant, thermal conductivity, and thermoelectric power of composite ceramic materials as a function of the volume content of nickel particles are revealed. Near the percolation threshold, the experimental results of the behavior of the electrical conductivity and static permittivity as a function of the nickel volume content in these materials differ from the dependences calculated within the framework of the percolation theory in that the curve of the dependence of the permittivity has the form similar to the electrical conductivity curve. The origin of this discrepancy is explained by the formation of a continuous spatial structure of tunnel-connected conductors.