Can you elaborate on this. You can't interact with most backgrounds anyway?
If you bake something, like AO or indirect lighning, it's going to look awesome in a static scene, but as soon as you start to move things around, it will break the illusion because, since it not in realtime, lightning (or AO, or reflections, or anything) won't change.
Take this The Last of Us screen
There is some nice lighting going on here, but its all baked, hence, everthing is static, and you can't move anything. Take the box on the right. You can see that the part your facing is being lit thanks to a bouncing light (because as far as I can tell, the sun is behind it, so without a bounce it would be completely in shadow), and the lightning is right in this particular situation, but move the box, and place it a few meters back, next to the metal door (Without baking the lightmap again) and the box will still look like as if it's still being lit from the old position.
This applies to everything, take AO, lets say you have a wall, you place a wardrobe next to it, bake the AO, then you move the wardrobe: you're left with a black silhouette (the AO) where the wardrobe was.
This slows down development, but more importantly limits the scene interactivity, because you wouldn't not be able to, say, blow up the wardrobe during gameplay, because you'd be left with the AO on the wall.
So you either don't use any kind of AO on dynamic objects, or you use SSAO/HBAO/HDAO...