Let's make it real simple. Obviously there is some chance that I can predict which card I will randomly select from a deck of playing cards - 1:52. I can make a prediction, select a card, and see if my prediction matches the results. We would expect that every once in a while, after every 52 tries, on the average, I will get it right. But, if I could do this every single time - if I guessed every card correctly - would you still claim it is chance? How about every other card? How about every fifth card? There is a reasonable expectation that once in awhile I will get it right. But if I get it right every time or within some limit, or even if I only get it right on 1:51 tries, or 1:51.9 tries, eventually we can rule out chance with high confidence, based on the number of trials.
I may even get lucky and guess every card correctly for some finite number of trials. But if my luck is nothing but chance, as we do more and more trials, my average success rate should approach a value of 1:52, exactly. If this wasn't true, we wouldn't have Las Vegas. In the end, given enough non-psychic customers,

, the house always wins. And we know this with high confidence, by the odds, and by the size of the sample. If a significant percentage of gamblers were able to use psychic abilities, to enhance their odds of winning to a significant degree, eventually this would be evident in the average house winnings over time, and Vegas would have a problem.
It also important to remember that, just as with Vegas, our card test doesn't require that we use the same person for each trial. We can use a different person for each trial, but the results should be the same. This is why we could in principle test for "coincidence" for large numbers of people that each only have a few, or one relevant experience. It doesn't require that only one person has many experiences that could be tested. If large numbers of people have similar experiences, assuming that we can properly define what we mean by "similar experiences" and then design a good test, in principle we could check to see if chance is sufficient to explain the experiences, or not, to a level of confidence determined by the sample size.