Children younger than 5½ years of age do not draw inferences based on the quantity of information expressed (e.g. the inference that ‘some of the animals are sleeping’ implies ‘not all the animals are sleeping’) at the rates that adults do. Explanations of this difficulty include limited processing resources, shallow lexical entries and lack of adult-like expectations of informativeness. I review experimental evidence and theoretical accounts of the development of the ability to draw these inferences with the aim (a) to highlight links between potentially related phenomena (such as these inferences and word learning), and (b) to outline a novel account which makes predictions about the underlying mechanisms and age of acquisition of these inferences.