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In this work, new achievable rates are derived for the uplink channel of a cellular network with joint multicell processing (MCP), where unlike previous results, the ideal backhaul network has finite capacity per cell. Namely, the cell sites are linked to the central joint processor via lossless links with finite capacity. The new rates are based on compress-and-forward schemes combined with local decoding. Further, the cellular network is abstracted by symmetric models, which render analytical treatment plausible. For this family of idealistic models, achievable rates are presented for both Gaussian and fading channels. The rates are given in closed form for the classical Wyner model and the soft-handover model. These rates are then demonstrated to be rather close to the optimal unlimited backhaul joint processing rates, even for modest backhaul capacities, supporting the potential gain offered by the joint MCP approach. Particular attention is also given to the low-signal-to-noise ratio (SNR) characterization of these rates through which the effect of the limited backhaul network is explicitly revealed. In addition, the rate at which the backhaul capacity should scale in order to maintain the original high-SNR characterization of an unlimited backhaul capacity system is found.