For purposes of this puzzle, a chess position consists of an arrangement of pieces on the board, as well as the side to move. No other information about the game state is included.

The objective of this puzzle is to identify a position that can occur twice in a single game but cannot occur three times.

More generally, what numbers $n$ are such that some position can occur $n$ times but not $n+1$ times?

For purposes of this puzzle, a chess position consists of an arrangement of pieces on the board, as well as the side to move. No other information about the game state is included.

The objective of this puzzle is to identify a position that can occur twice in a single game but cannot occur three times.

More generally, what numbers $n$ are such that some position can occur $n$ times but not $n+1$ times?

Note that we're using the FIDE ruleset, so 3-fold repetition and 50 move rules are optional for the players to call and basically irrelevant to this puzzle (but the 5-fold and 75 move rules are mandatory and may or may not be useful to a solution).