Q:

Is it correct to use "neither", "nor" and "but" together? 

Is this sentence grammatically and stylistically correct? 

They mightn't experience neither obsession nor attraction but real
  love. 

A:

I would suggest the sentence to be rewritten without the double negative:

They might experience neither obsession nor attraction but real love. 

You could even split it into two sentences, especially for spoken conversation:

What they might experience is neither obsession nor attraction.  It is real love.