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Modern machine learning applications are characterized by the increasing size of deep models and the growing diversity of data modalities. This trend underscores the importance of efficiently adapting pre-trained multi-modal models to the test distribution in real time, i.e., multi-modal test-time adaptation. In practice, the magnitudes of multi-modal shifts vary because multiple data sources interact with the impact factor in diverse manners. In this research, we investigate the the under-explored practical scenario *uni-modal distribution shift*, where the distribution shift influences only one modality, leaving the others unchanged. Through theoretical and empirical analyses, we demonstrate that the presence of such shift impedes multi-modal fusion and leads to the negative transfer phenomenon in existing test-time adaptation techniques. To flexibly combat this unique shift, we propose a selective adaptation schema that incorporates multiple modality-specific adapters to accommodate potential shifts and a ``router'' module that determines which modality requires adaptation. Finally, we validate the effectiveness of our proposed method through extensive experimental evaluations.Code available at https://github.com/chenmc1996/Uni-Modal-Distribution-Shift.