This study proposes a novel method for assessing the time-variant reliability of bridge structures by combining deep learning and regular vine (R-vine) copula models. The sample convolution and interaction network (SCINet) model, which is a deep learning algorithm, is established to predict future structural responses at different control-monitoring points. The R-vine copula model captures the complex correlations of the failure modes associated with different control-monitoring points and models their dependence structure. By integrating the SCINet and R-vine copula models, the SCINet-pair-copula model (SCINet-PCM) is established, and a framework based on SCINet-PCM is proposed to forecast the time-variant reliability of bridge structures. The results for a practical bridge reveal that the calculated results obtained using the proposed SCINet-PCMs satisfy the quality-inspection results of the bridge and are closer to the benchmark results of importance sampling, indicating that the proposed method is accurate and reasonable.