It has been conjectured that, for all $n$, there is no interval of length $n$ with more primes in it than the interval between $2$ and $n+1$. You look at a table of primes, and you see how they thin out the higher up you go, and that's evidence, of a sort. But more precisely, we know that the density of the primes among the first $n$ numbers goes to zero as $n$ goes to infinity, so that seems like a heuristic supporting the conjecture.
And the conjecture hasn't actually been disproved, but some 40 years ago, Hensley & Richards proved it contradicted the prime $k$-tuples conjecture, which has what's considered to be stronger supporting evidence. So at least one of two conjectures with heuristic support is false, we just don't have a decision yet on which one.
