Depending on what type of drink you are ordering, you could ask for a can (and they also bring a cup of ice). We always ask for a can (DH gets real thisty at altitude ). We have never been turned down.
One thing that I've noticed is when the flight attendants come around to pick up the trash (assuming you're not on a short flight and they are getting ready for landing), they almost always ask if folks would like anything else to drink. ^ So, if you are still thirsty when they come around to pick up the trash, I wouldn't hesitate to ask for another drink.