In this work, we examine the ability of NER models to use contextual information when predicting the type of an ambiguous entity. We introduce NRB, a new testbed carefully designed to diagnose Name Regularity Bias of NER models. Our results indicate that all state-of-the-art models we tested show such a bias; BERT fine-tuned models significantly outperforming feature-based (LSTM-CRF) ones on NRB, despite having comparable (sometimes lower) performance on standard benchmarks. To mitigate this bias, we propose a novel model-agnostic training method that adds learnable adversarial noise to some entity mentions, thus enforcing models to focus more strongly on the contextual signal, leading to significant gains on NRB. Combining it with two other training strategies, data augmentation and parameter freezing, leads to further gains.