Analytical derivation of bit error rates for multi-user coherent chaos-shift-keying (CSK) communication systems are presented in this paper. Nearly exact results are obtained by applying the central limit theorem of statistics to sums of independent variables. Based on χ2 distribution approximations, more viable but still very accurate results decrease complexity of the calculations. The χ2 approach is compared with the widely used Gaussian approximation approach to show its superiority in most cases. Bit error performance bounds for the multi-user CSK system from the approach are deduced as further contributions of this paper. The theoretical results obtained are entirely consistent with a range of simulations.