As entity type systems become richer and more fine-grained, we expect the number of types assigned to a given entity to increase. However, most fine-grained typing work has focused on datasets that exhibit a low degree of type multiplicity. In this paper, we consider the high-multiplicity regime inherent in data sources such as Wikipedia that have semi-open type systems. We introduce a set-prediction approach to this problem and show that our model outperforms unstructured baselines on a new Wikipedia-based fine-grained typing corpus.