In how many cases of K, P vs. K, N, N endings can the stronger side force checkmate? For instance, it wouldn't be possible if the Pawn was on the verge of queening.  But is it always possible if the Pawn started off on its original square?  Is it usually possible?  Or is it only rarely possible?  How is it accomplished?
Let's assume that white is the side with the two knights, and black has the single pawn. The endgame study composer A. A. Troitzky gave an analysis (in a lengthy supplement to his 1937 Collection of Chess Studies) that established what is now known as the "Troitzky line":



So long as a white knight has the black pawn securely blockaded on one of these squares (or further back), then white can win no matter the other arrangement of the pieces. Now there are also times when white can still win with the pawn already advanced beyond the Troitzky line, but then it usually depends on the location of the black king; that is, the board usually breaks down into a drawing region for the black king, and a losing region for the black king. Establishing that was also part of Troitzky's work.



I'd be way out of my depth trying to say anything about the details of Troitzky's analysis, and besides, he spent a good 60 pages or so working out the things that he did. But it's not too hard to get the conceptual idea behind the Troitzky line. The most basic winning approach for white is this: to blockade the pawn with one knight, then use the king and other knight to corner the black king, and finally release the pawn to move while bringing the blockading knight in to deliver checkmate.

In order for this to work, white needs a fortuitous combination of (1) the pawn not being too quick to queen, and (2) the white knight not being too slow to deliver checkmate. Since the knight is most mobile - has the most travel options - in the center of the board ("A knight on the rim is dim" and all that), it stands to reason that the further away from the center the knight starts, the more time it will need for a win to be guaranteed in general, and so the further back the pawn needs to be from queening. This reality is reflected pretty clearly in the inner V shape of the Troitzky line.

But what about those far advanced rook pawns in the Troitzky line? Well, as often happens with rook pawns, they lead to special cases. Here, the reason white has this additional leeway is because of Stamma's mate, as is pointed out, for instance, in section 35.03 of this article on ChessCafe by Karsten M&uuml;ller (of Fundamental Chess Endings fame).



One caveat in all this is that the Troitzky line does not take into account the 50-move rule; though there are wins in all those cases, some of them take longer than 50 moves to convert against perfect play. In the article I just mentioned, M&uuml;ller raised the question of determining a "second Troitzky line" that does take the 50-move rule into account. And in a followup article he announced that Helmut Conrady solved that problem by using Rafael Andrist's Wilhelm program to analyze the Nalimov tablebases. This second Troitzky line is



So if a white knight securely blockades the black pawn on one of these squares (or further back), then white can always win in under 50 moves. Note that while there's no guarantee of a 50-move win with either a b- or g-pawn, Conrady's analysis also showed that if the pawn is blockaded on b6/g6 or further back (as in the first Troitzky line), then for over 99% of the possible positions there is a win in under 50 moves.