Federated learning (FL) enables collaborative model training without exposing clients' private data, but its deployment is often constrained by the communication cost of transmitting gradients between clients and the central server, especially under system heterogeneity where low-bandwidth clients bottleneck overall performance. Lossy compression of gradient data can mitigate this overhead, and error-bounded lossy compression (EBLC) is particularly appealing for its fine-grained utility-compression tradeoff. However, existing EBLC methods (e.g., SZ), originally designed for smooth scientific data with strong spatial locality, rely on generic predictors such as Lorenzo and interpolation for entropy reduction to improve compression ratio. Gradient tensors, in contrast, exhibit low smoothness and weak spatial correlation, rendering these predictors ineffective and leading to poor compression ratios. To address this limitation, we propose an EBLC framework tailored for FL gradient data to achieve high compression ratios while preserving model accuracy. The core of it is an innovative prediction mechanism that exploits temporal correlations across FL training rounds and structural regularities within convolutional kernels to reduce residual entropy. The predictor is compatible with standard quantizers and entropy coders and comprises (1) a cross-round magnitude predictor based on a normalized exponential moving average, and (2) a sign predictor that leverages gradient oscillation and kernel-level sign consistency. Experiments show that this new EBLC yields up to 1.53x higher compression ratios than SZ3 with lower accuracy loss. Integrated into a real-world FL framework, APPFL, it reduces end-to-end communication time by 76.1%-96.2% under various constrained-bandwidth scenarios, demonstrating strong scalability for real-world FL deployments.