Due to structural effects, mechanically buckled tubes may exhibit negative stiffness (NS). The NS composites, formed by embedding the NS and positive-stiffness (PS) phases, have been shown to exhibit exceptional enhancements in their effective stiffness and viscoelastic damping. Through molecular dynamics simulation with the first-generation Tersoff–Brenner interatomic potential, we studied the composite system of the buckled (5,5) carbon nanotube (CNT), negative stiffness provider, being compressed laterally with a carbon fullerene (Cx) for x = 20, 60, 100, and 540. It was found the CNT-Cx system showed about 10 N/m for its spring constant in the linear range. When x = 20, 60, and 100, after reaching maximum loading, applied total force vs. lateral compression showed plateau-type behavior due to a series of local buckling, indicating it is a low-energy cost process to revert a buckled nanotube to an unbuckled state with the creation of local buckling. When x = 540, hardening-type force–displacement relationship was observed due to the interplay between the NS and PS phases. For all of the four cases, the effective stiffness along the loading direction was negative for large lateral compression since the buckled CNT provided more negative stiffness than the positive stiffness in the fullerene.