Surgical data science has seen rapid advancement with the excellent performance of end-to-end deep neural networks (DNNs). Despite their successes, DNNs have been proven susceptible to minor "corruptions," introducing a major concern for the translation of cutting-edge technology, especially in high-stakes scenarios. We introduce the SegSTRONG-C challenge dedicated to better understanding model deterioration under unforeseen but plausible non-adversarial "corruption" and the capabilities of contemporary methods that seek to improve it. Built on a dataset generated through counterfactual robotic replay, SegSTRONG-C provides paired clean and "corrupted" samples, enabling reproducible evaluation of model robustness. Participants are challenged to train tool segmentation algorithms on "uncorrupted" data and evaluate them on "corrupted" test domains for the binary robot tool segmentation task. Through comprehensive baseline experiments and participating submissions from widespread community engagement, SegSTRONG-C reveals key themes for model failure and identifies promising directions for improving robustness. The performance of challenge winners, achieving an average 0.9394 DSC and 0.9301 NSD across the unreleased test sets with "corruption" types: bleeding, smoke, and low brightness. This highlights how prior knowledge, customized training strategies, and architectural choice can be leveraged to improve robustness. In conclusion, the SegSTRONG-C challenge has identified practical approaches for enhancing model robustness. However, most approaches rely on conventional techniques that have known limitations. Looking ahead, we advocate for expanding intellectual diversity and creativity in non-adversarial robustness beyond data augmentation, calling for new paradigms that enhance universal robustness to unforeseen "corruptions" to facilitate richer applications in surgical data science.