I'm going to disagree with a lot of the moms here by saying I really hate it when we go to a party and the presents aren't opened there. DD is 4, and we have ALWAYS made a deal of giving presents--she goes to the store, picks out what she thinks her friend will like. We tell her the budget--usually $10-15--and if she picks out something too expensive, we tell her so, or if its not age appropriate. We see this as an important TEACHING opportunity for DD, showing her how to give gifts, think of others, etc. She gets REALLY excited about giving the gift to her friend, and has been really disappointed at parties where she doesn't get to see her friend open it. She interprets that as her friend not being interested in the gift--"why did she not open my present, mommy?" We try to explain, but I do feel like it is being ungrateful to deny the giver the pleasure of seeing their gift opened--particularly at such a young age, when we are trying to teach about generosity, sharing, etc.
I agree with the mom below that we need to set expectations for our kids, and help them to meet them. At DD's 4th b-day party, the gift opening was a little chaotic, but DH and I were in the midst, helping kids take turns giving gifts, making sure DD said "thank you" after each one, etc. The kids were all REALLY excited to actually GIVE their gift to DD--as DD is when she goes to their parties. Again, this is all a teaching moment for our kids.
Honestly, DD doesn't need more toys. B-day present giving is NOT about getting stuff, nor should it be. It's about the experience of giving and receiving, learning how to do that thoughtfully and gratefully. All of that is learned through actually giving and opening the gifts, not through packing everything in the trunk to take home for later.
And on a further note, this can start WAY EARLY. We started at DD's 2nd Christmas, when she was 20 mos old. She already "got" it--she made ornaments for family members, physically gave them to the recipients, watched them open them, then helped put them on the tree. You could see how proud she was to participate in the gift exchanging. Another important lesson--Christmas is about GIVING, not receiving. And also not about competition--a handmade ornament gets as much excitement as a $50 toy. Really, it's all about what we want to teach our kids, and what birthday parties are supposed to really be about.