Open-set recognition (OSR) methods aim to identify whether or not a test example belongs to a category observed during training. Depending on how visually similar a test example is to the training categories, the OSR task can be easy or extremely challenging. However, the vast majority of previous work has studied OSR in the presence of large, coarse-grained semantic shifts. In contrast, many real-world problems are inherently fine-grained, which means that test examples may be highly visually similar to the training categories. Motivated by this observation, we investigate three aspects of OSR: label granularity, similarity between the open- and closed-sets, and the role of hierarchical supervision during training. To study these dimensions, we curate new open-set splits of a large fine-grained visual categorization dataset. Our analysis results in several interesting findings, including: (i) the best OSR method to use is heavily dependent on the degree of semantic shift present, and (ii) hierarchical representation learning can improve coarse-grained OSR, but has little effect on fine-grained OSR performance. To further enhance fine-grained OSR performance, we propose a hierarchy-adversarial learning method to discourage hierarchical structure in the representation space, which results in a perhaps counter-intuitive behaviour, and a relative improvement in fine-grained OSR of up to 2% in AUROC and 7% in AUPR over standard training. Code and data are available: langnico.github.io/fine-grained-osr.