The mechanical instabilities and viscoelastic response of individual multiwalled carbon nanotubes and nanofibers (MWCNTs/Fs) under uniaxial compression are studied with atomic force microscopy. Specific buckling events are evident by regimes of negative stiffness, i.e., marked drops in force with increasing compression. Uniaxial cyclic loading can be repeatedly executed even in initially postbuckled regimes, where the CNTs/Fs display incremental negative stiffness. Increases in mechanical damping of 145–600 % in these initially postbuckled regimes, as compared to the linear prebuckled regimes, are observed. Increased damping is attributed to frictional energy dissipation of walls in buckled configurations of the MWCNTs/Fs. This represents the extension of the concept of negative stiffness to the scale of nanostructures and opens up possibilities for designing nanocomposites with high stiffness and high damping simultaneously.