I typed 'No, I don't' as a translation of 'Nee, ik niet', which was accepted. The alternative (more correct?) translation is 'No, not me.'
In English these are two quite different sentences which sometimes work in a similar way but not always.
For example, if I asked 'Are you next?', the answer 'no, not me.' would be good, but 'no, I don't' wouldn't be.
So, how does 'Nee, ik niet' fit into that difference?
"Bent u de volgende?" (Are you next?) - You can answer this with "No, not me." or "No, I am not." in English and with "Nee, ik niet." in Dutch.
"Zwem jij?" (Do you swim?) - You can answer this with "No, I do not." in English and with "Nee, ik niet." in Dutch.
The difference between Dutch and English is that in Dutch you don't have to use a "do" verb with negation. In English you do, depending on what verb is being used. As you pointed out, "No, I don't." doesn't always work. "Nee, ik niet." works for all (implied) verbs.
Hope I'm making sense here :)
I'm guessing (though I didn't try) that "No, I'm not" also works. It's probably just used more generally, because it doesn't have a verb in it, so it can apply in more situations. Literally it translates as "No, I not". So it splits the difference, really :)
I just tried "No, I'm not" and it worked. so I guess it really could be used for many situations