Modern LLMs can now produce highly readable abstractive summaries, to the point where traditional automated metrics for evaluating summary quality, such as ROUGE, have become saturated. However, LLMs still sometimes introduce unwanted content into summaries, i.e., information inconsistent with or unsupported by their source. Measuring the occurrence of these often subtle ``hallucinations'' automatically has proved to be challenging. This in turn has motivated development of a variety of metrics intended to measure the factual consistency of generated summaries against their source. But are these approaches measuring what they purport to do? In this work, we stress-test automatic factuality metrics. Specifically, we investigate whether and to what degree superficial attributes of summary texts suffice to predict ``factuality'', finding that a (supervised) model using only such shallow features is reasonably competitive with SOTA factuality scoring methods. We then evaluate how factuality metrics respond to factual corrections in inconsistent summaries and find that only a few show meaningful improvements. In contrast, some metrics are more sensitive to benign, non-factual edits. Motivated by these insights, we show that one can ``game'' (most) automatic factuality metrics, i.e., reliably inflate ``factuality'' scores by appending innocuous sentences to generated summaries.Taken together, our results raise questions about the degree to which we should rely on existing automated factuality metrics and what exactly we want ``factuality metrics'' to measure.