Why not 4.Bd2 in the Nimzo-Indian? After the initial moves of the Nimzo-Indian 4.Bd2 is rarely played and it gets poor results according to databases.

[FEN ""]
1. d4 Nf6 2. c4 e6 3. Nc3 Bb4 4. Bd2


It looks like the most natural move to me, because:


After Bxc3 Bxc3 White keeps the bishop pair and preserves the pawn structure.
It breaks the pin, so the knight can influence the center again.  
It doesn't block any piece and it develops a piece.  


And Bd2 is played against Bb4 in other openings.

So, why the move is weak?
There are really two questions here.

Why is 4.Bd2 rarely played?

In a nutshell, 4.Bd2 is playable, but there are better alternatives.


  After Bxc3 Bxc3 White keeps the bishop pair and preserves the pawn structure.


Even if Black decides to reply with 4...Bxc3, White won't keep his bishop pair after 5...Ne4.

[FEN ""]
1. d4 Nf6 c4 e6 Nc3 Bb4 Bd2 Bxc3 Bxc3 Ne4 Qc2 Nxc3 Qxc3
(6. Bb4 c5 dxc5 Qf6 Nf3 Qxb2 Rb1 Qxa2)



  It breaks the pin, so the knight can influence the center again.


Black pins the knight to prevent White from playing e4. Even after 4.Bd2, White cannot do this.

[FEN ""]
1. d4 Nf6 c4 e6 Nc3 Bb4 Bd2 O-O e4? Bxc3 Bxc3 Nxe4



  It doesn't block any piece and it develops a piece.


But it leaves the pawn on d4 undefended and prevents the queen from defending the bishop if it develops to d3.

Let's look at the alternatives:

[FEN ""]
1. d4 Nf6 c4 e6 Nc3 Bb4 e3
(4. Qc2)


4.e3 (Rubinstein System) ignores the pin, prepares to develop the bishop to d3, protects the pawn on d4 and takes White one step closer to castling.

4.Qc2 (Classical Variation) is a better alternative to 4.Bd2 if you want to avoid doubled pawns at all costs.

Why does 4.Bd2 get poor results according to databases?

As I said before, 4.Bd2 is playable. In fact, it's what Stockfish suggests after 3...Bb4. However, engines are particularly bad at openings and, as we have seen before, there are better alternatives.

Therefore, 4.Bd2 is mostly played by amateurs, which face the Nimzo-Indian and do not know the corresponding theory.

This makes the winning percentages in databases entirely useless. For example, according to 365Chess.com, White's first move with the highest winning percentage is 1.Na3.