The masses and radii of low-magnetic field neutron stars can be measured by combining different observable quantities obtained from their X-ray spectra during thermonuclear X-ray bursts. One of these quantities is the apparent radius of each neutron star as inferred from the X-ray flux and spectral temperature measured during the cooling tails of bursts, when the thermonuclear flash is believed to have engulfed the entire star. In this paper, we analyze 13,095 X-ray spectra of 446 X-ray bursts observed from 12 sources in order to assess possible systematic effects in the measurements of the apparent radii of neutron stars. We first show that the vast majority of the observed X-ray spectra are consistent with blackbody functions to within a few percent. We find that most X-ray bursts follow a very well determined correlation between X-ray flux and temperature, which is consistent with the whole neutron-star surface emitting uniformly during the cooling tails. We develop a Bayesian Gaussian mixture algorithm to measure the apparent radii of the neutron stars in these sources, while detecting and excluding a small number of X-ray bursts that show irregular cooling behavior. This algorithm also provides us with a quantitative measure of the systematic uncertainties in the measurements. We find that those errors in the spectroscopic determination of neutron-star radii that are introduced by systematic effects in the cooling tails of X-ray bursts are in the range similar or equal to 3%-8%. Such small errors are adequate to distinguish between different equations of state provided that uncertainties in the distance to each source and the absolute calibration of X-ray detectors do not dominate the error budget.