Language comprehenders routinely make pragmatic inferences that go beyond the literal meanings of utterances. If A said “I ate some of the cookies,” B should infer that A ate some but not all. Children perform poorly on experimental tests of scalar implicatures like this, despite their early-emerging sensitivity to pragmatic cues. Our current work explores factors responsible for children’s successes and failures for computing pragmatic inferences. In two experiments, we used eye-tracking to whether children compute implicatures when given access to contextual alternatives to the target word (Experiment 1), and when they hear prosodic cues that emphasize the contrast between the target and alternative (Experiment 2). We found that four-year-olds successfully identify the inferential referent in this paradigm; with supportive prosodic cues, we saw evidence of success in three-year-olds as well. In sum, with sufficient contextual support, preschool children are capable of making online pragmatic inferences.