Scanning tunneling microscopy measurements of tunneling through molecules adsorbed on a surface have been simulated using a standard empirical model based upon the Wentzel-Kramer-Brillouin method applied to tunneling through a barrier. The Gaussian noise inherent in these experiments has been added to the model data using a Monte Carlo technique. By generating multiple sets of current-voltage curves and fitting these to the model we have evaluated how reliably barrier height can be determined as a function of noise level. The results suggest that for constant percentage standard deviation in the noise greater than 5% the barrier height cannot be determined reliably. At this level, the standard deviation in the estimate of the barrier height is about 10%. Weighted fits give more reliable estimates of the barrier height. If the height of the tip above the molecule is known, so that the fit is only a single parameter the barrier height can be determined reliably even at percentage noise levels as high as 20%. However, in this case unweighted fits must be used otherwise the estimated value deviates by up to 15% from the true value. Data with constant absolute noise give similar results. The effects of experimental resolution have been evaluated in a similar manner and are shown to have a significant influence on the reliability. At a resolution of about 0.1% of full scale the standard deviation in the estimate of barrier height is only about 2% but increases rapidly to 10% for a resolution of about 1%.