Channel state information (CSI) is of vital importance in wireless communication systems. Existing CSI acquisition methods usually rely on pilot transmissions, and geographically separated base stations (BSs) with non-correlated CSI need to be assigned with orthogonal pilots which occupy excessive system resources. Our previous work adopts a data-driven deep learning based approach which leverages the CSI at a local BS to infer the CSI remotely, however the relevance of CSI between separated BSs is not specified explicitly. In this paper, we exploit a model-based methodology to derive the Cramér-Rao lower bound (CRLB) of remote CSI inference given the local CSI. Although the model is simplified, the derived CRLB explicitly illustrates the relationship between the inference performance and several key system parameters, e.g., terminal distance and antenna array size. In particular, it shows that by leveraging multiple local BSs, the inference error exhibits a larger power-law decay rate (w.r.t. number of antennas), compared with a single local BS; this explains and validates our findings in evaluating the deep-neural-network-based (DNN-based) CSI inference. We further improve on the DNN-based method by employing dropout and deeper networks, and show an inference performance of approximately $90\%$ accuracy in a realistic scenario with CSI generated by a ray-tracing simulator.