The birefringence induced by partially ordered polymer networks dispersed in an isotropic phase of liquid crystals is studied. Polymer networks were formed by the polymerization of 1-4 % mixtures of prepolymer in the nematic phase of liquid crystals. The partial nematic (paranematic) ordering is analyzed in terms of the Landau-de Gennes approach using a simple model of an array of thin fibrils, which can, on a large scale, form bundles of polymer-rich material. The comparison of the theory and birefringence data clearly shows that the main building bloc of the network is a few nanometers thick fibril. The average thickness of fibrils varies slightly with the polymer concentration. The information obtained on large scale (micrometer) structures is not conclusive and must be complemented by other methods. The network on the average retains 30-50 % of the order parameter of the environment where it was formed.