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](https://eudml.org/doc/205282), Acta Arith 25 (1973/74) 375-391, [MR0396440](https://mathscinet.ams.org/mathscinet-getitem?mr=0396440)) 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).