Is it possible to animate a soft body object? What I'm trying to do is create a few bouncing jello cubes. I can make a jello cube. And I can scale it to make it bounce. I can't do both.
Scaling it requires Soft Body Goal to be enabled, but enabling Soft Body Goal glues it in place and prevents it from actually bouncing around the scene.  

Bring in more of these objects, and you have another issue in that they don't want to collide. You can sort of fix that last issue by tacking on a Collision modifier but it seems a bit buggy. And I'm also back to not being able to scale the cube. As user3800527 suggested I tried tacking on a Rigid Body to that, but that just led to some really weird behavior.

As far as trying to make the soft body animateable, I've tried just plain animating the scale of the object as well as animating the Soft Body Edge Spring Length property, but those don't appear to affect the simulation while it's taking place. Soft Body Goal is able to slightly fix this, but when you turn that on you're trading that for the soft body's freedom of movement.

Here's an example blend file showing some of what I've tried so far, layers 1-3 have different things I've tried, and all of them have the bottom cube scale from 1 to 2 from frame 20 to 23. http://pasteall.org/blend/index.php?id=43836

So is something like this even currently possible in blender?
Yes this is possible

First create a floor cube large wide but not that high.
Make it a Passive rigid body.  (so it wont fall down its the floor)

Next a bit above it add a cube, subdivide it a few times.
First add rigid body collision (convex hull, source deform).
Next add softbody to this cube.

It will drop and woble on the floor.

oh and for better results use damping on soft body.
such cubes can bounche on eachother as well if you copy them shift D