To generalize Hugh Thomas' answer, one option might be to take a look at what the Bonferonni Inequalities give you. Essentially you can stop inclusion-exclusion after any subtraction and you'll always be left with a lower bound.
So if you don't have enough of the Nk to run inclusion-exclusion all the way through, or if it makes the computation intractable, see what the best lower bound you can get from what you have is.