In this paper, we study network-coded cooperative diversity (NCCD) systems comprising multiple sources, one relay, and one destination. The relay detects the packets received from all sources and performs Galois field network coding. We propose a simple cooperative maximum-ratio combining scheme for the destination which is shown to achieve the maximum diversity gain of the system. Furthermore, we provide a mathematical framework for the asymptotic analysis of NCCD systems with M-ary modulation for high signal-to-noise ratios. Based on this framework, we derive simple and elegant closed-form expressions for the asymptotic symbol and bit error rates which provide significant insight into the impact of various system and channel parameters on performance and can be exploited for performance optimization. Simulation results confirm the accuracy of the presented analysis and show that large performance gains are possible by optimizing the power allocation in NCCD systems based on the developed analytical results.