Generic statements convey generalizations about categories, but how generic predications combine is unclear. “Elephants live in Africa and Asia” does not mean that individual elephants live on both continents. In addition, such conjunctive generics pose interesting questions for theories of incremental processing because the meaning of the sentence can change part-way through: “Elephants live in Africa” would imply most or all do, but “Africa and Asia” implies some live in each. We extend a recently proposed computational model of generic language understanding with an incremental processing mechanism that can begin to interpret an utterance before a speaker has finished their sentence. This model makes novel predictions about partial interpretations of conjunctive generic sentences, which we test in two behavioral experiments. The results support a strong view of incrementality, wherein listeners continuously update their beliefs based on expectations about where a speaker will go next with their utterance.