Inspired by the dexterous skill of human hands, we studied the mechanics of fingers when exerting forces with the tip. We show that the finger is prone to buckle under compressive forces and the underlying mechanics resemble the bucking of a slender column. This raises the question of why healthy human hands rarely suffer such buckling instabilities. To investigate neural feedback control and the role of muscle viscoelasticity, we examined the stability of the index finger when human volunteers pushed on rigid surfaces with their fingertip. We analyzed high-speed video of rare buckling events in human volunteers and examined the limits of force production by externally stabilizing the finger to test the hypothesis that the neural system may limit maximal force production in order to avoid finger buckling. Through these experiments and mathematical analyses of a neuromuscular model, we show that the index finger is stabilized by the active elastic response of muscles and not by neural feedback control. Our findings show the important role played by the elastic response of muscles in grasping. Furthermore, these findings have implications for the design and control of compliant, yet stable, robotic manipulators.