We explore the impact of potential growth of data breaks within GNSS position time series on the uncertainty of site velocity estimate, which is important for many geodetic and geophysical applications. Based on a linear regression model consisting of linear rate, annual sinusoids, and breaks, the rate uncertainty is estimated for decade-long time series of various errors, including white noise, fractal white noise, flicker noise, first-order- Gauss-Markov noise, random walk, fractal random walk, and mixture of flicker noise and white noise. For each error model, the sensitivity of rate uncertainty to the number of data breaks is numerically assessed by computing the inflation rate of uncertainty as the number of data breaks increases. We show that the sensitivity depends on the degree of temporal correlation within the error. For uncorrelated white noise, each additional data break increases the rate uncertainty by 100% of its value when the time series is free of data breaks; In contrast, for random walk noise, the increased occurrence of data breaks barely affects the rate uncertainty estimate. In addition, the sensitivity decays rapidly as the degree of error correlation increases. For the mixture of flicker noise plus white noise, which best describes the error of the majority of current GNSS position time series, the rate uncertainty increases by 11% when the number of data breaks doubles from one to two, much smaller than the value of 40% concluded by previous study. Therefore, given current noise characteristics of GNSS site position estimate, data break has very limited impact on the uncertainty of site velocity estimate. For the improvement of reference frame stability, the existence of data breaks is not a major obstacle before the temporal correlation of position error could be reduced sufficiently.