To achieve language proficiency, infants must find the building blocks of speech and master the rules governing their legal combinations. However, these problems are linked: words are also built according to rules. Here, we explored early morphosyntactic sensitivity by testing when and how infants could find either words or within-word structure in artificial speech snippets embodying properties of morphological constructions. We show that 12-month olds use statistical relationships between syllables to extract words from continuous streams, but find word-internal regularities only if the streams are segmented. Seven-month-olds fail both tasks. Thus, 12-month-olds infants possess the resources to analyze the internal composition of words if the speech contains segmentation information. However, 7-month-old infants may not possess them, although they can track several statistical relations. This developmental difference suggests that morphosyntactic sensitivity may require computational resources extending beyond the detection of simple statistics.