Make armature bend across multiple vertices

Question

My file:

I'm very, very new to Blender and I'm trying to rig up a really simple low-poly character. I've got it rigged but no matter what I do, the joints are only pivoting across one vertex, even though I gave it more vertices to bend on. No matter how I adjust the weights it's not bending the way I want.

I want this armature to bend across all three of the circled vertices equally, but it's only choosing one vertex to bend along and the others stay straight, causing clipping when bent too far.

Here's how it's bending now:

Here's my weights:

I've tried painting the weights every which way but it doesn't seem to change anything at all.

Answer 1

As we can see if we select one of these vertices it is assigned to several vertex groups (not sure what are the 2 firsts) but the weight of arm.l is much higher than the weight of shoulder.l, so the influence of the forearm will be much higher when you'll bend the arm:

What you could do is remove all this edge loop from any group:

Then assign it to the groups called shoulder.l and arm.l with the same Weight (it can be 1, as long as the weight will be the same, both the 2 bones will have the same influence):

Also maybe check the adjacent edge loop, and enable the Preserve Volume option of the Armature modifier. Result:

Also you could try some alternative topologies:

Or:

If you want the vertices of solution 1 to merge so that you get solution 2, select all these vertices, assign them to a new vertex group that you'll call Weld for example:

Give your object a Weld modifier, select the Weld vertex group and increase the Distance until the vertices merge when the arm is bent: