Fermat's last theorem was known for regular primes (Kummer, 1850). It was conjectured that there are infinitely many regular primes (Siegel 1964), moreover that a majority of primes (about 65%) are regular. For any specific prime, one can check whether it is regular or not, and there are tables of them up to 25000.

Thus it is likely, but not proved, that Kummer established FLT for infinitely many primes.