grammar - The meaning of しないでもなかった

彼の性格が読めてきたような気がしないでもなかった

しない is a negative conjugation of する

And from what I understand でもなかった is the equivalent of "was not"

So it's a double denial? How does this work?

Answer

Yes, しないでもない (≒しなくもない） is a double-negative construction, which can be seen as a set phrase used to say something less confidently, or to reluctantly admit something.

彼の性格が読めてきたような気がしないでもなかった

(lit.) It was not that I didn't feel I came to be able to read his character.
I feel I have gradually learnt how to understand his character.

Similar examples:

• 失敗だったと言えなくもない。　≒　失敗だったと言えないでもない。
  (lit.) It is not that one cannot say it was a failure.
  One could argue that it was a failure.


• 100万円はなくもない。　≒　100万円はないでもない。
  
  (lit.) It is not that I don't have one million yen.
  (Admittedly,) I have one million yen.