Initial searching in Japanese-Japanese dictionaries seem to suggest that it's along the lines of "has a bit of ..., so saying otherwise would be wrong", "more so than not", and is a synonym to ないわけではない (though I personally don't really understand ないわけではない either).
Am I understanding it correctly? What would be an accurate and/or good way (these might not be the same) to translate it to English when it's part of a sentence, or is it something that's heavily context-dependent? Would it be something like "it's not that [negative clause]"?
While I do have a specific example of it being used that I'm interested in, I'm also interested in the general case.
P.S., A fan translation of that example translated it as "Unfortunately yes". Is this a good translation? Does the expression itself convey the exasperation, or is it simply in the tone it's said?