State-of-the-art pre-trained language models have been shown to memorise facts and perform well with limited amounts of training data. To gain a better understanding of how these models learn, we study their generalisation and memorisation capabilities in noisy and low-resource scenarios. We find that the training of these models is almost unaffected by label noise and that it is possible to reach near-optimal results even on extremely noisy datasets. However, our experiments also show that they mainly learn from high-frequency patterns and largely fail when tested on low-resource tasks such as few-shot learning and rare entity recognition. To mitigate such limitations, we propose an extension based on prototypical networks that improves performance in low-resource named entity recognition tasks.