Prior to age four, children succeed in non-elicited-response false-belief tasks but fail elicited-response false-belief tasks. To explain this discrepancy, the processing-load account argues that the capacity to represent beliefs emerges in infancy, as indicated by early success on non-elicited-response tasks, but that children's ability to demonstrate this capacity depends on the processing demands of the task and children's processing skills. When processing demands exceed young children's processing abilities, such as in standard elicited-response tasks, children fail despite their capacity to represent beliefs. Support for this account comes from recent evidence that reducing processing demands improves young children's performance: when demands are sufficiently reduced, 2.5-year-olds succeed in elicited-response tasks. Here we sought complementary evidence for the processing-load account by examining whether increasing processing demands impeded children's performance in a non-elicited-response task. 3-year-olds were tested in a preferential-looking task in which they heard a change-of-location false-belief story accompanied by a picture book; across children, we manipulated the amount of linguistic ambiguity in the story. The final page of the book showed two images: one that was consistent with the main character's false belief and one that was consistent with reality. When the story was relatively unambiguous, children looked reliably longer at the false-belief-consistent image, successfully demonstrating their false-belief understanding. When the story was ambiguous, however, this undermined children's performance: looking times to the belief-consistent image were correlated with verbal ability, and only children with verbal skills in the upper quartile of the sample demonstrated a significant preference for the belief-consistent image. These results support the processing-load account by demonstrating that regardless of whether a task involves an elicited response, children's performance depends on the processing demands of the task and their processing skills. These findings also have implications for alternative, deflationary accounts of early false-belief understanding.