We analyze the role of the force-dependent kinetics of motor proteins in the stability of antiparallel arrays of polar filaments, such as those in the mitotic spindle. We determine the possible stable structures and show that there exists an instability associated to the collective behavior of motors that leads to the collapse of the structure. The agreement of our results and several experimental observations in eukaryotic cell division suggests an important role of kinesin-5 motors and microtubule bundles in the stability of the mitotic spindle.