Cascading failures (CF) entail component breakdowns spreading through infrastructure networks, causing system-wide collapse. Predicting CFs is of great importance for infrastructure stability and urban function. Despite extensive research on CFs in single networks such as electricity and road networks, interdependencies among diverse infrastructures remain overlooked, and capturing intra-infrastructure CF dynamics amid complex evolutions poses challenges. To address these gaps, we introduce the \textbf{I}ntegrated \textbf{I}nterdependent \textbf{I}nfrastructure CF model ($I^3$), designed to capture CF dynamics both within and across infrastructures. $I^3$ employs a dual GAE with global pooling for intra-infrastructure dynamics and a heterogeneous graph for inter-infrastructure interactions. An initial node enhancement pre-training strategy mitigates GCN-induced over-smoothing. Experiments demonstrate $I^3$ achieves a 31.94\% in terms of AUC, 18.03\% in terms of Precision, 29.17\% in terms of Recall, 22.73\% in terms of F1-score boost in predicting infrastructure failures, and a 28.52\% reduction in terms of RMSE for cascade volume forecasts compared to leading models. It accurately pinpoints phase transitions in interconnected and singular networks, rectifying biases in models tailored for singular networks. Access the code at https://github.com/tsinghua-fib-lab/Icube.