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The application of distributed model predictive controllers (DMPC) for multi-agent systems (MASs) necessitates communication between agents, yet the consequence of communication data rates is typically overlooked. This work focuses on developing stability-guaranteed control methods for MASs with limited data rates. Initially, a distributed optimization algorithm with dynamic quantization is considered for solving the DMPC problem. Due to the limited data rate, the optimization process suffers from inexact iterations caused by quantization noise and premature termination, leading to sub-optimal solutions. In response, we propose a novel real-time DMPC framework with a quantization refinement scheme that updates the quantization parameters on-line so that both the quantization noise and the optimization sub-optimality decrease asymptotically. To facilitate the stability analysis, we treat the sub-optimally controlled MAS, the quantization refinement scheme, and the optimization process as three interconnected subsystems. The cyclic-small-gain theorem is used to derive sufficient conditions on the quantization parameters for guaranteeing the stability of the system under a limited data rate. Finally, the proposed algorithm and theoretical findings are demonstrated in a multi-AUV formation control example.