Use of Ni and Ni when "either/or" or "neither/nor" are not used in English translation

The lesson covers this in the introduction :  Whereas in English you have three ways to express the negation - not either… or / neither… nor… / not... or... - in French, you only use ne... ni... ni... 

There  are multiple versions with the same basic meaning in English but they all translate the same way in French ‘ne..ni..ni’. In reverse, the single version in French can be translated to any of the English versions and be correct. Even if one sounds like a more direct translation, that does not mean it is ‘more’ correct.

The same principle applies with soit .. soit in French - this is frequently used when the English does not use ‘either .. or’,  but uses just ‘or’. 

Merci Maarten,

Thank you for your detailed explanation.  I still have some questions. 

The test question stated: "Il a vu Paul et Sam ? -Non, ________ .", with no translation.

If a translation of "No, he hasn't seen Paul OR Sam", (Or one of the other variations, "He has seen neither Paul nor Sam."; "He hasn't seen either Paul or Sam."), had been given, then my response would have used one of the choices outlined. But, I mistook the test to be about using a negative with the passe compose rather than referring to the use of "ne...ni...ni". 

So, my question now: 

Is this an incorrect sentence in French? : "Il n'a pas vu Paul et Sam." 

That is, would you always be required to say "Il n'a vu ni Paul ni Sam." ? Which to me is stating something different, because the first sentence implies that "Paul and Sam" are together, while in the second they are not necessarily together. 

For example:

"Mon mari est alle en ville, mais il n'a pas vu Paul et Sam au marche comme d'habitude." (In this case, Paul and Sam always go to the market together.)

"Mon mari est alle en ville, mais il n'a vu ni Paul ni Sam au marche." (Here, Paul and Sam are not necessarily associated with one another. They may be known to the subject of the sentence, "mon mari" but they might not even know each other.)

I appreciate your patience with this.

I don’t think this is so much a question of correct grammar but typical usage and expression in French. French is often more precise, and assumptions made in English are often not made in the same manner. In this case, ne .. ni. ..ni .. makes clear ‘not seeing either of them’ (therefore not seeing ‘both of them’).

In the situation you describe of ‘being a couple’, in English ‘not seeing the couple’, leaves open the question of whether one or other was seen, and the assumption is made that if there is no clarification to that effect, neither was seen. The typical French response is unambiguous in the first instance. Although, in everyday speech with context known, ‘Non, il ne les a pas vus’ is likely to be heard.

Sometimes the answer ‘expected’ is because of the quiz and lesson it is attached to, but in this case, I think the response anticipated is just what would be common in French.