We've never had a problem with extending or adding on on any of our trips. I've only seen like one person on any board say they were told no due to some particular circumstance. As long as you are paying for the extra costs out of your pocket, it should almost never be a problem, it's not really changing anything on the sponsor's end.
We've gone on 5 trips and have 2 more upcoming ones, and we always pay whatever extra expenses there are to bring our children--mainly the extra seats on the plane. Every sponsor said it was no problem.