An important question today is whether a given text was used to train a large language model (LLM). A \emph{completion} test is often employed: check if the LLM completes a sufficiently complex text. This, however, requires a ground-truth definition of membership; most commonly, it is defined as a member based on the $n$-gram overlap between the target text and any text in the dataset. In this work, we demonstrate that this $n$-gram based membership definition can be effectively gamed. We study scenarios where sequences are \emph{non-members} for a given $n$ and we find that completion tests still succeed. We find many natural cases of this phenomenon by retraining LLMs from scratch after removing all training samples that were completed; these cases include exact duplicates, near-duplicates, and even short overlaps. They showcase that it is difficult to find a single viable choice of $n$ for membership definitions. Using these insights, we design adversarial datasets that can cause a given target sequence to be completed without containing it, for any reasonable choice of $n$. Our findings highlight the inadequacy of $n$-gram membership, suggesting membership definitions fail to account for auxiliary information available to the training algorithm.