This paper examines the success of various Markov-chain models of daily precipitation series in reproducing the characteristics of area-average rainfall in Britain. The first model considered is the standard twos-tate first-order Markov renewal process coupled to an amount model using the incomplete G-probability distribution. We find that variability of seasonal totals and autocorrelation of daily amounts are both too small in this model, compared with observations. These are serious deficiencies, often overlooked, and possibly related. We proceed to consider models involving Markov chains of higher (temporal) order and many states, both of which generalizations may increase autocorrelation. A second-order two-state model is no better than the first-order, but a first-order many-state model captures a high fraction of the seasonal variability, because use of many states improves the model's representation of spells of heavy precipitation, which appear to have a considerable influence on the seasonal variance. Better still is a second-order many-state model, a type which, to our knowledge, has not previously been investigated. We suggest that the best model would have a continuum of states, rather than a discrete set. Our conclusion is that a large proportion of seasonal variability may be explained in terms of the average daily structure, but there may be a residual component caused by processes operating on longer time-scales and possibly predictable with reference to these. Reproduction of long-period (e.g. monthly or seasonal) variance and of the structure of daily autocorrelation provide crucial tests of stochastic "weather generators", and we recommend that models which fail to simulate these statistics realistically be used only with great caution.