Yes, to highest order the fractional error (percentage of the outcomes the error represents) goes as the inverse square root of the number of samples. Barring other factors such as experimental design and representation of the samples, you're usually hitting around 1% error at 100 samples.

One doesn't actually need a ton of samples to get good statistical significance depending on the design and results of the study. Of course, there are factors that can increase or decrease these rates depending on if you are getting a little Bayesian or not