I actually just had to do this a few days ago on a necklace I got for Christmas. It was a pendant that came with a 16" chain. I wanted something longer so I went back to Tiffany's.
The SA said that she could lengthen it for me at a cost of $15 per inch. That would take a couple of weeks. Or I could buy the 18" chain for $50 or the 20" one for $65. I ended up buying the 20" chain.