The question is, what is "enough times" to prove the 20% out? It's not going to be measured in 100's or even 1,000's. Any statisticians care to answer that?
It depends on how certain you want to be. The uncertainty depends not on the number of tries, but on the number of successes. With 100 successes, one standard deviation is about 10 (or 10%). With 1000 successes, it is around 31.6 (3%). More generally, it goes as roughly the square roof of the number of successes. If you know how many standard deviations away you are from the expected value, you can calculate the probability of the expectation being valid.
This is in fact exactly what Darth Sweets did. He had 400-odd tries (i.e. with a 20% chance, the expectation is 80) and 53 successes so he is a little over 3 standard deviations away. He did the calculation and it works out to about a 3 in a 1000 chance of the RE probability being 20%. Like a lot of people have said earlier in the thread, this is improbable, but hardly impossible.
By the way, I tried to record my RE attempts and what I got at the point when I stopped was 26 successes out of 125 attempts (i.e. it's basically dead-on 20%, even with a relatively large uncertainty).