This paper presents an experimental investigation of nanocomposites composed of three ratios of epoxy/graphene nanoplatelets (GNPs) by weight. The 0.1, 0.2, and 0.3 wt.% specimens were carefully manufactured, and their mechanical and thermal conductivity properties were examined. The tensile strength and modulus of epoxy/GNPs were enhanced by the large surface area of graphene nanoplatelets, causing crack deflection that created new fracture fronts and friction because of the rough fracture surface. However, the compressive strength was gradually reduced as GNP loading percentages increased. This was probably due to severe plastic yielding on the epoxy, leading to catastrophic axial splitting caused by premature fractures. Furthermore, the highest thermal conductivity was 0.1283 W/m-K, representing a 20.92% improvement over neat epoxy (0.1061 W/m-K) when 0.3 wt.% GNPs were added to the epoxy. This was because of efficient heat propagation in the GNPs due to electron movement through percolative paths. The tensile failure mode in epoxy/GNP nanocomposites showed a few deflected and bifurcated rough cracks and brittle, dimple-like fractures. Contrarily, compressive failure mode in GNP-added epoxy showed plastic flexural buckling and brittle large-axial splitting. The epoxy/GNP nanocomposites were considered a damage-tolerant material.