Color constancy estimates illuminant chromaticity to correct color-biased images. Recently, Deep Neural Network-driven Color Constancy (DNNCC) models have made substantial advancements. Nevertheless, the potential risks in DNNCC due to the vulnerability of deep neural networks have not yet been explored. In this paper, we conduct the first investigation into the impact of a key factor in color constancy-brightness-on DNNCC from a robustness perspective. Our evaluation reveals that several mainstream DNNCC models exhibit high sensitivity to brightness despite their focus on chromaticity estimation. This sheds light on a potential limitation of existing DNNCC models: their sensitivity to brightness may hinder performance given the widespread brightness variations in real-world datasets. From the insights of our analysis, we propose a simple yet effective brightness robustness enhancement strategy for DNNCC models, termed BRE. The core of BRE is built upon the adaptive step-size adversarial brightness augmentation technique, which identifies high-risk brightness variation and generates augmented images via explicit brightness adjustment. Subsequently, BRE develops a brightness-robustness-aware model optimization strategy that integrates adversarial brightness training and brightness contrastive loss, significantly bolstering the brightness robustness of DNNCC models. BRE is hyperparameter-free and can be integrated into existing DNNCC models, without incurring additional overhead during the testing phase. Experiments on two public color constancy datasets-ColorChecker and Cube+-demonstrate that the proposed BRE consistently enhances the illuminant estimation performance of existing DNNCC models, reducing the estimation error by an average of 5.04% across six mainstream DNNCC models, underscoring the critical role of enhancing brightness robustness in these models.