Information about the position of an object that is held in both hands, such as a golf club or a tennis racquet, is transmitted to the human central nervous system from peripheral sensors in both left and right arms. How does the brain combine these two sources of information? Using a robot to move participant's passive limbs, we performed psychophysical estimates of proprioceptive function for each limb independently and again when subjects grasped the robot handle with both arms. We compared empirical estimates of bimanual proprioception to several models from the sensory integration literature: some that propose a combination of signals from the left and right arms (such as a Bayesian maximum-likelihood estimate), and some that propose using unimanual signals alone. Our results are consistent with the hypothesis that the nervous system both has knowledge of and uses the limb with the best proprioceptive acuity for bimanual proprioception. Surprisingly, a Bayesian model that postulates optimal combination of sensory signals could not predict empirically observed bimanual acuity. These findings suggest that while the central nervous system seems to have information about the relative sensory acuity of each limb, it uses this information in a rather rudimentary fashion, essentially ignoring information from the less reliable limb.