Time-series of precipitable water vapour (PWV), derived from continuous Global Positioning System (GPS) observations, are analysed for the two South African stations HRAO and SUTH. Since water vapour is a major greenhouse gas, observed changes in atmospheric PWV could be indicative of weather and climate change. Our main contribution is a realistic noise model of the PWV observations which enables one to draw correct conclusions about the significance of the derived PWV increase or decrease for given time spans longer than five years. It is demonstrated that the PWV residuals that are obtained after fitting a trend and yearly signal to the data are, due to the simple model's exclusion of short-term scatter, much larger than the PWV uncertainties provided by the GPS analysis software. Although a better solution for the associated uncertainties is obtained by using the variance of these PWV residuals for the uncertainty rescaling, it is shown that the ARMA(1,1) noise model better represents the associated statistical uncertainties than the simple white noise model. The ARMA(1,1)-derived PWV trend uncertainties are approximately 2 times greater than those for a rescaled white noise model. Finally, it is argued that the variability of the annual signal prevents any trend estimation using time series shorter than about five years. A quantitative measure is presented to determine the minimum period of continuous GPS observational data required to measure PWV trends to a specified accuracy. As result of our study, we conclude that no statistically significant PWV trends are observed at the two GPS stations between 1998 and 2006.