This paper presents an experimental study to investigate how humans interact with a robotic arm simulating primarily unstable, damping-defined, mechanical environments, and to quantify lower bounds of robotic damping that humans can stably interact with. Human subjects performed posture maintenance tasks while a robotic arm simulated a range of negative damping-defined environments and transiently perturbed the human arm to challenge postural stability. Analysis of 2-dimensional kinematic responses in both the time domain and phase space allowed us to evaluate stability of the coupled human-robot system in both anterior-posterior (AP) and medial-lateral (ML) directions, and to determine the lower bounds of robotic damping for stable physical human-robot interaction (pHRI). All subjects demonstrated higher capacity to stabilize their arm against negative damping-defined environments in the AP direction than the ML direction, evidenced by all 3 stability measures used in this study. Further, the lower bound of robotic damping for stable pHRI was more than 3.5 times lower in the AP direction than the ML direction: -30.0 Ns/m and -8.2 Ns/m in the AP and ML directions, respectively. Sensitivity analysis confirmed that the results in this study were relatively insensitive to varying experimental conditions. Outcomes of this study would allow us to design a less conservative robotic impedance controller that utilizes a wide range of robotic damping, including negative damping, and achieves more transparent and agile operations without compromising coupled stability and safety of the human-robot system, and thus improves the overall performance of pHRI.