Pólya’s conjecture is the usual example of a statement that’s true for many cases, but breaks down for some large value of n. In that example, n=906150257. However, I think I have a new favourite: gcd(n^17 + 9, (n+1)^17 + 9) = 1 for all n < 8424432925592889329288197322308900672459420460792433 (More counterexamples at https://oeis.org/A010034)