A long-standing conjecture in Number Theory is that for each positive integer $n$ there is no stretch of $n$ consecutive integers containing more primes than the stretch from 2 to $n+1$. Just looking at a table of primes and seeing how they thin out is enough to make the conjecture plausible.
But Hensley and Richards (Primes in intervals, Acta Arith 25 (1973/74) 375-391, MR0396440) proved that this conjecture is incompatible with an equally long-standing conjecture, the prime $k$-tuples conjecture.
The current consensus, I believe, is that prime $k$-tuples is true, while the first conjecture is false (but not proved to be false).