Nearly uncoupled Markov chains (aka nearly completely decomposable Markov chains) arise in a variety of applications, where they model loosely coupled systems. In such systems it may be difficult to determine the transitions probabilities with high accuracy. This paper investigates the sensitivity of the limiting distribution of the chain to perturbations in the transition probabilities. The conclusion is that nearly uncoupled Markov chains are quite sensitive to such perturbations but the perturbation of the limiting distribution is not arbitrary. (Also cross-referenced as UMIACS-TR-90-18) Appeared in Numerical Solutions of Markov Chains, W. J. Stewart ed., Dekker, New York, 1990.