4...O-O followed by 6...Ne8!? is a popular line in the f3 Nimzo at master level, but I'm still struggling to understand why it's played.
[FEN ""] 1.d4 Nf6 2.c4 e6 3.Nc3 Bb4 4.f3 O-O 5.a3 Bxc3 6.bxc3 Ne8!?
In this video, Caleb Denby seems to imply the main reason is Black wants to go after the c4-pawn with a typical Nc6-Na5 / b6-Ba6 plan, but wants to stop an eventual pin with Bg5 that could be annoying for Black and "force him to make concessions in the center." But why not just throw in the move ...h6 at some point instead? eg.
[FEN ""] 1. d4 Nf6 2. c4 e6 3. Nc3 Bb4 4. f3 O-O 5. a3 Bxc3+ 6. bxc3 c5 7. e4 d6 8. Bd3 Nc6 9. Ne2 h6
To me, it seems pretty bizarre to use a full move to undevelop your knight to a bad square and potentially let White Greek gift you in the future, in anticipation of a move that White isn't even likely to play for another 3 turns.