To better understand the developmental trajectory of children's pragmatic development, studies that examine more than one type of implicature as well as associated linguistic and cognitive factors are required. We investigated three- to five-year-old English-speaking children's (N = 71) performance in ad hoc quantity, scalar quantity and relevance implicatures, as well as word learning by exclusion inferences, using a sentence-to-picture-matching story-based task. Children's pragmatic abilities improved with age, with word learning by exclusion acquired first, followed by relevance and ad hoc quantity implicatures, and finally scalar quantity implicatures. In an exploratory analysis (with a subset of the data N = 58), we found that structural language knowledge was a predictor of pragmatic performance (but no evidence for an association with socioeconomic status or Theory of Mind, controlling for structural language). We discuss reasons why this developmental pattern emerges with reference to linguistic and extra-linguistic properties of these inferences.