Google for sample size calculators.
To achieve a 95% confidence level with a 5% or smaller margin of error, for a population of 2000 students, you’d need at least 323 samples. If you keep that confidence and MOE constant, the required sample size doesn’t increase too much over college-sized populations. But if your surveys of T20 colleges are getting results that are overlapping within the margin of error (and your goal is to differentiate them), you’d want to decrease the MOE, hence increase the number of samples, right? To get a 3% MOE for a population of 10K undergrads you’d need 965 survey samples.