The detection of hyperbole is an important stepping stone to understanding the intentions of a hyperbolic utterance. We propose a model that combines pre-trained language models with privileged information for the task of hyperbole detection. We also introduce a suite of behavioural tests to probe the capabilities of hyperbole detection models across a range of hyperbole types. Our experiments show that our model improves upon baseline models on an existing hyperbole detection dataset. Probing experiments combined with analysis using local linear approximations (LIME) show that our model excels at detecting one particular type of hyperbole. Further, we discover that our experiments highlight annotation artifacts introduced through the process of literal paraphrasing of hyperbole. These annotation artifacts are likely to be a roadblock to further improvements in hyperbole detection.