Note that Nicely's website lists far more prime gaps than only the numbers 4 to 2000 that Tony Huynh mentioned, but in separate tables: Tables of prime gaps

He doesn't seem to mention up to which number all even gaps are known to occur; from manual inspection it seems that all even numbers up to 52,400 occur as gaps between two consecutive proven primes. (52,402 is only listed as the gap between two probable primes, and so are several later entries). Above 100,000, there are several even numbers which aren't even listed as gaps between probable primes, e.g. 116,254.

Note that Nicely's website lists far more prime gaps than only the numbers 4 to 2000 that Tony Huynh mentioned, but in separate tables: Tables of prime gaps (Wayback Machine)

He doesn't seem to mention up to which number all even gaps are known to occur; from manual inspection it seems that all even numbers up to 52,400 occur as gaps between two consecutive proven primes. (52,402 is only listed as the gap between two probable primes, and so are several later entries). Above 100,000, there are several even numbers which aren't even listed as gaps between probable primes, e.g. 116,254.