Efficient heat dissipation from modern electronic devices is a key issue for their proper performance. An important role in the assembly of electronic devices is played by polymers, due to their simple application and easiness of processing. The thermal conductivity of pure polymers is relatively low and addition of thermally conductive particles into polymer matrix is the method to enhance the overall thermal conductivity of the composite. The aim of the presented work is to examine a possibility of increasing the thermal conductivity of the filled epoxy resin systems, applicable for electrical insulation, by the use of composites filled with graphene nanoplatelets. It is remarkable that the addition of only 4 wt.% of graphene could lead to 132 % increase in thermal conductivity. In this study, several new aspects of graphene composites such as sedimentation effects or temperature dependence of thermal conductivity have been presented. The thermal conductivity results were also compared with the newest model. The obtained results show potential for application of the graphene nanocomposites for electrical insulation with enhanced thermal conductivity. This paper also presents and discusses the unique temperature dependencies of thermal conductivity in a wide temperature range, significant for full understanding thermal transport mechanisms.