Cross-modal hashing (CMH) has appeared as a popular technique for cross-modal retrieval due to its low storage cost and high computational efficiency in large-scale data. Most existing methods implicitly assume that multi-modal data is correctly labeled, which is expensive and even unattainable due to the inevitable imperfect annotations (i.e., noisy labels) in real-world scenarios. Inspired by human cognitive learning, a few methods introduce self-paced learning (SPL) to gradually train the model from easy to hard samples, which is often used to mitigate the effects of feature noise or outliers. It is a less-touched problem that how to utilize SPL to alleviate the misleading of noisy labels on the hash model. To tackle this problem, we propose a new cognitive cross-modal retrieval method called Robust Self-paced Hashing with Noisy Labels (RSHNL), which can mimic the human cognitive process to identify the noise while embracing robustness against noisy labels. Specifically, we first propose a contrastive hashing learning (CHL) scheme to improve multi-modal consistency, thereby reducing the inherent semantic gap. Afterward, we propose center aggregation learning (CAL) to mitigate the intra-class variations. Finally, we propose Noise-tolerance Self-paced Hashing (NSH) that dynamically estimates the learning difficulty for each instance and distinguishes noisy labels through the difficulty level. For all estimated clean pairs, we further adopt a self-paced regularizer to gradually learn hash codes from easy to hard. Extensive experiments demonstrate that the proposed RSHNL performs remarkably well over the state-of-the-art CMH methods.