In supervised machine learning, privileged information (PI) is information that is unavailable at inference, but is accessible during training time. Research on learning using privileged information (LUPI) aims to transfer the knowledge captured in PI onto a model that can perform inference without PI. It seems that this extra bit of information ought to make the resulting model better. However, finding conclusive theoretical or empirical evidence that supports the ability to transfer knowledge using PI has been challenging. In this paper, we critically examine the assumptions underlying existing theoretical analyses and argue that there is little theoretical justification for when LUPI should work. We analyze LUPI methods and reveal that apparent improvements in empirical risk of existing research may not directly result from PI. Instead, these improvements often stem from dataset anomalies or modifications in model design misguidedly attributed to PI. Our experiments for a wide variety of application domains further demonstrate that state-of-the-art LUPI approaches fail to effectively transfer knowledge from PI.