Human communication is based on a variety of inferences that we draw from sentences, often going beyond what is literally said. While there is wide agreement on the basic distinction between entailment, implicature, and presupposition, the status of many inferences remains controversial. In this paper, we focus on three inferences of plain and embedded disjunctions, and compare them with regular scalar implicatures. We investigate this comparison from the novel perspective of the predictions of state-of-the-art large language models, using the same experimental paradigms as recent studies investigating the same inferences with humans. The results of our best performing models mostly align with those of humans, both in the large differences we find between those inferences and implicatures, as well as in fine-grained distinctions among different aspects of those inferences.