In a negative sentence, you use negative pronouns and adverbs, too. But those don't cancel out, nor do they add in any way . So you get things like:
I don't know anything => Nic nevím (lit. I don't know nothing)
Nobody has ever done this yet => Nikdo to ještě nikdy neudělal (lit.: nobody hasn't never done this yet)
I've never did anything wrong to anyone => Nikdy jsem nikomu neudělal nic špatného. (lit. I haven't never done nothing wrong to no one.).
(It's probably the same way in Russian. The Slavonic languages can be a bit tricky... )