Membership Inference attacks (MIAs) aim to predict whether a data sample was present in the training data of a machine learning model or not, and are widely used for assessing the privacy risks of language models. Most existing attacks rely on the observation that models tend to assign higher probabilities to their training samples than non-training points. However, simple thresholding of the model score in isolation tends to lead to high false-positive rates as it does not account for the intrinsic complexity of a sample. Recent work has demonstrated that reference-based attacks which compare model scores to those obtained from a reference model trained on similar data can substantially improve the performance of MIAs. However, in order to train reference models, attacks of this kind make the strong and arguably unrealistic assumption that an adversary has access to samples closely resembling the original training data. Therefore, we investigate their performance in more realistic scenarios and find that they are highly fragile in relation to the data distribution used to train reference models. To investigate whether this fragility provides a layer of safety, we propose and evaluate neighbourhood attacks, which compare model scores for a given sample to scores of synthetically generated neighbour texts and therefore eliminate the need for access to the training data distribution. We show that, in addition to being competitive with reference-based attacks that have perfect knowledge about the training data distribution, our attack clearly outperforms existing reference-free attacks as well as reference-based attacks with imperfect knowledge, which demonstrates the need for a reevaluation of the threat model of adversarial attacks.