Learning structural information from observational data is central to producing new knowledge outside the training corpus. This holds for mechanistic understanding in scientific discovery as well as flexible test-time compositional generation. We thus study how language models learn abstract structures and utilize the learnt structural information at test-time. To ensure a controlled setup, we design a natural language dataset based on linguistic structural transformations. We empirically show that the emergence of learning structural information correlates with complex reasoning tasks, and that the ability to perform test-time compositional generation remains limited.