The Gilbert model (1-st order Markov chain model) and the single-multiplexer model are two frequently used models in the study of packet-loss processes in communication networks. In this paper we investigate the accuracy of the Gilbert model, and higher-order Markov chain extended Gilbert models, in characterizing the packet-loss process associated with a transport network modeled in terms of a single-multiplexer. More specifically, we quantitatively compare the packet-loss statistics predicted by the Gilbert models with those predicted by an exact queueing analysis of the single-multiplexer model. This topic is important since low-complexity Gilbert models are frequently used to characterize end-to-end network packet-loss behavior. On the other hand, network congestion behavior is often characterized in terms of a single bottleneck node modeled as a multiplexer. It is of some interest then to establish the relative accuracy of Gilbert models in predicting the packet-loss behavior on even such a simplified network model. We demonstrate that the Gilbert models have some serious deficiencies in accurately predicting the packet-loss statistics of the single-multiplexer model. The results are shown to have some serious consequences for the performance evaluation of forward error correction (FEC) coding schemes used to combat the effects of packet losses due to network buffer overflows.