Zhang's theorem and Polignac's conjecture
Zhang proved that any admissible set of $k_0=3\,500\,000$ (or more) numbers contains some $a,b$ such that $n+a$ and $n+b$ are both prime infinitely often. There is an admissible set of this size with values between 0 and 70,000,000 thus the statement that there are infinitely many prime gaps at most 70 million.
The best value proved for $k_0$ so far is 50, which leads to the gap of 246 via the admissible tuple (0, 4, 6, 16, 30, 34, 36, 46, 48, 58, 60, 64, 70, 78, 84, 88, 90, 94, 100, 106, 108, 114, 118, 126, 130, 136, 144, 148, 150, 156, 160, 168, 174, 178, 184, 190, 196, 198, 204, 210, 214, 216, 220, 226, 228, 234, 238, 240, 244, 246).
But if you wanted you could choose a different tuple which showed, for example, that there are infinitely many prime gaps in a different range. For example, the admissible 50-tuple (0, 4, 10, 16, 22, 30, 34, 42, 46, 52, 60, 64, 70, 76, 84, 90, 94, 100, 106, 112, 126, 130, 136, 142, 150, 154, 160, 172, 184, 192, 202, 210, 214, 220, 226, 232, 240, 244, 252, 262, 270, 276, 280, 286, 294, 312, 316, 324, 330, 336) proves that there are infinitely many prime gaps of length between 4 and 336 (inclusive).*
So if the current methods were extended to prove the twin prime conjecture it would automatically prove Polignac's conjecture. Now that might be too much to expect -- the Polymath project has already changed its methodology in significant ways in the course of its several months of operation. But it does serve to show that Polignac's conjecture is not far from the twin prime conjecture.
A reasonable question, then, is "can Zhang's method be so extended?". At the moment the answer seems to be "no": even on the assumption of the generalized Elliott-Halberstam conjecture, the best that has been achieved is $k_0=3$ which means (via the 3-tuple (0, 2, 6)) that at least one of twin primes, cousin primes, and sexy primes have infinitely many members. But even with that high-powered assumption we can't narrow it down further.
* Similarly I can show that there are infinitely many prime gaps between 6 and 378, between 8 and 502, between 10 and 616, between 12 and 678, and so forth. On GEH the best you can do is $g$ to $2g$ if $3|g$ or $g$ to $2g+2$ otherwise.