Neural image compression (NIC) is increasingly used in computer vision pipelines, as learning-based models are able to surpass traditional algorithms in compression efficiency. However, learned codecs can be unstable and vulnerable to adversarial attacks: small perturbations may cause severe reconstruction artifacts or indirectly break downstream models. Despite these risks, most NIC benchmarks only emphasize rate-distortion (RD) performance, focusing on model efficiency in safe, non-adversarial scenarios, while NIC robustness studies cover only specific codecs and attacks. To fill this gap, we introduce \textbf{NIC-RobustBench}, an open-source benchmark and evaluation framework for adversarial robustness of NIC methods. The benchmark integrates 8 attacks, 9 defense strategies, standard RD metrics, a large and extensible set of codecs, and tools for assessing both the robustness of the compression model and impact on downstream tasks. Using NIC-RobustBench, we provide a broad empirical study of modern NICs and defenses in adversarial scenarios, highlighting failure modes, least and most resilient architectures, and other insights into NIC robustness. Our code is available online at https://github.com/msu-video-group/NIC-RobustBench.