(edit: As Reid Barton pointed out I'm assuming here that you have some sort of lower bound on the Nk as well as an upper bound...if this is not the case then including more terms won't help at all)
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.