(edit: As Reid Barton pointed out I'm assuming here that you have some sort of lower bound on the N<sub>k</sub> 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][1]  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 N<sub>k</sub> 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.


  [1]: http://en.wikipedia.org/wiki/Boole%27s_inequality