There is a surprising degree of overlapping structure evident across the languages of the world. One factor leading to cross-linguistic similarities may be constraints on human learning abilities. Linguistic structures that are easier for infants to learn should predominate in human languages. If correct, then (a) human infants should more readily acquire structures that are consistent with the form of natural language, whereas (b) non-human primates' patterns of learning should be less tightly linked to the structure of human languages. Prior experiments have not directly compared laboratory-based learning of grammatical structures by human infants and non-human primates, especially under comparable testing conditions and with similar materials. Five experiments with 12-month-old human infants and adult cotton-top tamarin monkeys addressed these predictions, employing comparable methods (familiarization-discrimination) and materials. Infants rapidly acquired complex grammatical structures by using statistically predictive patterns, failing to learn structures that lacked such patterns. In contrast, the tamarins only exploited predictive patterns when learning relatively simple grammatical structures. Infant learning abilities may serve both to facilitate natural language acquisition and to impose constraints on the structure of human languages.