〜ないでもない
sort of, somewhat (can see, feel)
Verb ない-form + でもない
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ないでもない is a hesitant double negative meaning "I do, a little": その気持ちも分からないでもない, "I can sort of understand how you feel".
It softens an admission, as if you're reluctant to agree fully. It's common with verbs of understanding, feeling and seeing: 分かる, 理解できる, 気がする, 似ている, 考える.
It's close to ないことはない (N3), and ないものでもない is the most formal of the three.
Don't confuse it with ないで, "without". Here でもない is "it's not that it isn't". The effect is a grudging, partial agreement, which is useful when you want to be fair to someone's view without fully endorsing it.
Examples
その気持ちも分からないでもない。
I can sort of understand how you feel.
彼の言うことも理解できないでもない。
I can see where he's coming from, to some extent.
少し寂しい気がしないでもない。
I do feel a little lonely, I suppose.
言われてみれば、似ていないでもない。
Now that you mention it, they do look a bit alike.
転職を考えないでもないが、まだ決めていない。
I have sort of been thinking about changing jobs, but I haven't decided.