in over 1500 hours of playing Civ5, yes I had every AI as my opponent. And yes some of them expand. But they are a tiny minority.
Once I wanted to play a 'huge empires' game. So I set one up using the following settings: continents, large, legendary start, epic, 10 CS, opponents usa, russia, china, germany. I quit that game when I realized that the one guy on my continent, washington, had 2 citys when I had like 10. There was a lot of valuable land, he just never settled. I thought that maybe it was the barbs fault, so I restarted a game with the same setting, but 'no barbs' as well and therefor replacing germany with rome. And guess what? I got russia as my landmass-buddy and she grew to three citys and stopped.
I am realy surprised to read that there are players here who have never encountered that. For me theres allways huge spaces of free land. Its so annoying. This realy breaks the game for me. I cant see any reason for colonization or wars while theres still room. It just seems dumb.