We investigated thermal properties of the epoxy-based composites with the high loading fractionup to f ≈ 45 vol %of the randomly oriented electrically conductive graphene fillers and electrically insulating boron nitride fillers. It was found that both types of the composites revealed a distinctive thermal percolation threshold at the loading fraction fT > 20 vol %. The graphene loading required for achieving thermal percolation, fT , was substantially higher than the loading, fE , for electrical percolation. Graphene fillers outperformed boron nitride fillers in the thermal conductivity enhancement. It was established that thermal transport in composites with high filler loadings, f ≥ fT , is dominated by heat conduction via the network of percolating fillers. Unexpectedly, we determined that the thermal transport properties of the high loading composites were influenced strongly by the cross-plane thermal conductivity of the quasi-two-dimensional fillers. The obtained results shed light on the debated mechanism of the thermal percolation, and facilitate the development of the next generation of the efficient thermal interface materials for electronic applications.