As robots become increasingly prevalent in human environments, there will inevitably be times when the robot needs to interrupt a human to initiate an interaction. Our work introduces the first interruptibility-aware mobile-robot system, which uses social and contextual cues online to accurately determine when to interrupt a person. We evaluate multiple non-temporal and temporal models on the interruptibility classification task, and show that a variant of Conditional Random Fields (CRFs), the Latent-Dynamic CRF, is the most robust, accurate, and appropriate model for use on our system. Additionally, we evaluate different classification features and show that the observed demeanor of a person can help in interruptibility classification; but in the presence of detection noise, robust detection of object labels as a visual cue to the interruption context can improve interruptibility estimates. Finally, we deploy our system in a large-scale user study to understand the effects of interruptibility-awareness on human-task performance, robot-task performance, and on human interpretation of the robot's social aptitude. Our results show that while participants are able to maintain task performance, even in the presence of interruptions, interruptibility-awareness improves the robot's task performance and improves participant social perceptions of the robot.