fredthebear: <26...h6? indicates that Nimzowitsch did not even see 27.Nxe6! coming>
I will hazard to disagree. I believe Nimzowitsch probably knew he would be defeated with proper play at that point which is precisely why he makes the move 26...h6. This move has merit (when no other move solves Black's problems anyway).
Consider an aggressive move that was not played, that being 26.Qh5. It seems that 26.Qh5 h6 wins quickly for White, following up w/a knight sacrifice/offer instead of a queen sacrifice and she delivers checkmate on the 7th. The only useful defense to 26.Qh5 is 26...Qb1+ seizing the open line to scurry the Black queen back to defend the castle. Perhaps White saw this too and played the timid-looking 26.Kh2 instead to prepare a lethal Qh5 next. Now Black sees that Qb1 does not check to gain a tempo and White will have time to clog the diagonal by Rc2 or e4.
So, Black plays 26...h6 to discourage Qh5 from being played as the meddling knight would be lost. Qh5 has been prevented. If Black is really lucky White retreats the Ng5-Nf3 leaving Black with hopes of eventually winning! It's certainly worth a try in a desperate situation, but White stayed on course by offering the knight sacrifice anyway. Excellent knight move, although choices were limited.
Now Black makes three consecutive moves with his king's rook that looks absurd on the face of it, but there is method to the madness, even though White has a won game at this point.
After 27.Nxe6!, it makes sense for Black to save his threatened rook and try to exchange off both rooks on the c-file, defenders for attackers, to take the steam out of the attack. (At this point, Black has no legitimate counter play available and is at White's mercy, hoping White can be dissuaded from his mating net.) White knows better than to trade rooks; he wants to maintain at least one rook on the seventh rank. After 28.Qf3 Rf8 makes sense as well, willing to give up the exchange but w/chances of survival if White will accept.
Then comes 29.Qg3 g6. Black probably could have resigned here, but many players overlook queen sacrifices, so why give White the benefit of the doubt? Especially when White could have played the tempting Qg4 a move earlier but didn't -- possibly suggesting White is know playing move-to-move instead of having the mating net sequence clear in his head. (29.Qg3 attacking f7 is better than 29.Qg4 attacking g7, so White is not to be faulted for a second consecutive queen move to the g-file but there is an awkward look to playing Qg3 in two moves when Qg4 in one move has the same threat.)
Make your lesser opponent prove OTB that he can see the finish clearly and will not settle for a less than ideal move. The point of Black hanging on when he may have seen mate coming is that White could have succumbed to temptation and hastily exchanged off an attacking piece and/or retreated the en prise knight, etc. leaving Black w/a playable game. To White's credit, it did not happen in this game, but it certainly is worth a try on Black's part. (Furthermore, Nimzowitsch had a much different attitude toward resignation than did Schlechter.)
Even after 31...Kh8 Black has the slimmest of hopes of White being lured into making a greedy capture (knight takes rook or rook takes pawn) instead of ignoring both opportunities and doubling rooks on the seventh. Why resign when there is bait on the table that your opponent might just be foolish enough to take? Even though Nimzowitsch likely saw the mating net himself, he was right to play on in hopes that White would relax and abort the king hunt for material gain. Many a game was been ruined by making a greedy, ill-timed capture.
Right or wrong, these are my own thoughts while studying this finish. I did not refer to any publications regarding this game.