We have never opened the gifts at the party itself. Usually the kids are too busy playing and running around. Just more fun to let them play. We usually have the parties between 11-2 and then open the gifts up that afternoon after the kids have napped (my son is 3 1/2, my daughter will be 2 in 2 weeks :)). That way I can still get thank-you's out in a timely manner. I am sure as they get older, it will change, but it has worked for us. Mine also tend to get overwhelmed with too many toys, so usually the night of the party (or Christmas night, whenever the kids have a lot of gifts) I will take some and put them up in a bin out in the garage. Then as they get bored with things I will take some out.
ETA: We have tried opening the gifts up at the party, but it was more for the parents than the kids. The other kids were running around and playing on the playscape and bounce house, my son (at his 3rd birthday party) just wanted to be playing with his friends, so getting him to open them was a nightmare.