Recent earthquake recordings from high-rate (1 Hz) continuous GPS stations in California indicate that even higher temporal resolution provided by modern GPS receivers is desirable. We measured seven baselines during time intervals devoid of detectable transient signals at sampling rates of 1--50 Hz with geodetic receivers from four manufacturers to investigate the noise characteristics of these data. Our tests over short distances (meters) to typical station spacing (tens of kilometers) of regional GPS networks show no loss of spatial resolution compared to 1-Hz samples. Measurement noise is red with the typical ramp profile of log-log spectra below about 0.5 Hz. Above this frequency, noise is essentially white. Low-pass filtering of high-rate positions achieves improved spatial resolution compared to decimated raw samples. Averaging 20-Hz measurements to 2-Hz samples on a 40-km baseline, for example, yields about 0.5 mm horizontal and about 3--4 mm vertical accuracy at high frequencies. These estimates are a factor of 2--2.5 better than for 2-Hz raw samples. The improvements in spatial resolution due to averaging at high frequencies are substantial and approach the theoretical "square-root-of-n" expectation for independent samples. However, since noise spectral densities rise rapidly below about 0.5-Hz, low-pass filtering is only effective above this frequency. These results have important implications for the design of continuous GPS networks for crustal deformation and structural monitoring and for positioning and attitude determination of dynamic platforms.