negation - Do native speakers use double negatives in order to mean positive situations REALLY

I stumbled upon VOA (Voice Of America)'s video about the use of the combination of either/or and neither/nor.

Well, the guy in the video says, the double negatives, as he says, the sentence such as

Neither Adam nor his friend cannot swim.

But I rarely or never met a person this kind of double negatives spoke in order to mean the positive situations.

I know for sure that double negatives should mean positive, but how rare in the real word is is? Or am I dead wrong?

---

Thank you for your answer, JavaLatle,

However, will you kindly "break into parts" of your example according to which such double negatives turned positives can be used particularly in formal situation so that I can understand the English 1819's parliament quote? The part, especially after had they ~ is making difficult for me to grasp what this decree or statement is trying to say.

The executive government assuredly would not not have done its duty, had they not made use of all lawful means to thwart the designs of such confederacies... The parliamentary history of England, 1819