Advances in computational methods and big data availability have recently translated into breakthroughs in AI applications. With successes in bottom-up challenges partially overshadowing shortcomings, the 'human-like' performance of Large Language Models has raised the question of how linguistic performance is achieved by algorithms. Given systematic shortcomings in generalization across many AI systems, in this work we ask whether linguistic performance is indeed guided by language knowledge in Large Language Models. To this end, we prompt GPT-3 with a grammaticality judgement task and comprehension questions on less frequent constructions that are thus unlikely to form part of Large Language Models' training data. These included grammatical 'illusions', semantic anomalies, complex nested hierarchies and self-embeddings. GPT-3 failed for every prompt but one, often offering answers that show a critical lack of understanding even of high-frequency words used in these less frequent grammatical constructions. The present work sheds light on the boundaries of the alleged AI human-like linguistic competence and argues that, far from human-like, the next-word prediction abilities of LLMs may face issues of robustness, when pushed beyond training data.