This is about recent work on a longstanding and famous open question in number theory, called the 'twin prime conjecture'. The conjecture states: there are infinitely many pairs of prime numbers just two apart (e.g., 17 and 19 are twin primes, as 19 - 17 = 2). For a few centuries there has not been great (direct) progress on this problem, but in April, Zitang Zhang proved there are infinitely many pairs of primes at most 70 million apart. This breakthrough got everyone really excited, and the linked page shows the assault now being made by the mathematical community on the 70 million limit, using all kinds of techniques based on Zhang's argument, trying to drive the limit downwards, closer and closer to the dream of two.
No, arbitrarily large gaps between prime numbers exist. But there are no efficient methods for locating gaps of a particular length. The largest gap for which the end primes are known, so far, is of length 337,446.
The recent work summarized on the linked page proves something else, namely that there are infinitely many pairs of prime numbers separated by only 388,284. The goal is to get that number down to two and to prove the twin prime conjecture. (In your interpretation this would be like saying every prime is within 2 of another prime, which is obviously false.)
