In this paper, we present a framework that unites obstacle avoidance and deliberate physical interaction for robotic manipulators. As humans and robots begin to coexist in work and household environments, pure collision avoidance is insufficient, as human-robot contact is inevitable and, in some situations, desired. Our work enables manipulators to anticipate, detect, and act on contact. To achieve this, we allow limited deviation from the robot's original trajectory through velocity reduction and motion restrictions. Then, if contact occurs, a robot can detect it and maneuver based on a novel dynamic contact thresholding algorithm. The core contribution of this work is dynamic contact thresholding, which allows a manipulator with onboard proximity sensors to track nearby objects and reduce contact forces in anticipation of a collision. Our framework elicits natural behavior during physical human-robot interaction. We evaluate our system on a variety of scenarios using the Franka Emika Panda robot arm; collectively, our results demonstrate that our contribution is not only able to avoid and react on contact, but also anticipate it.