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In this paper, we show that for haptic rendering using position feedback, the structure of the feedback loop imposes a fundamental tradeoff between accurate rendering of virtual environments and sensitivity of closed-loop responses to hardware variations and uncertainty. Due to this tradeoff, any feedback design that achieves high-fidelity rendering incurs a quantifiable cost in terms of sensitivity. Analysis of the tradeoff reveals certain combinations of virtual environment and haptic device dynamics for which performance is achieved only by accepting very poor sensitivity. This analysis may be used to show that certain design specifications are infeasible and may guide the choice of hardware to mitigate the tradeoff severity. We illustrate the predicted consequences of the tradeoff with an experimental study.