How to create a "Rounded Grids" effect like in MagicaVoxel I can export a 3D model from MagicaVoxel as an OBJ file and import it into Blender no problems. The colours come UV-mapped onto a 256x1px PNG file.

I'm trying to get Blender to render this effect where each 'voxel' in the model has distinct rounded edges, like this:



I created a bump map image like so:



and then added a new texture to my material.

I set the mapping to "Generated" instead of UV. I set this to a Cube projection.



I got the scale right by setting the Texture Space like so:



Now when this new texture's Influence is set to Diffuse Color, I see this:



so the positioning seems correct.

However, when I set the influence to Geometry Normal, I see this:



What am I missing here? I guess this is because the Normal Map uses the RGB values to set the normals so I'm missing a dimension of displacement? If I try to use "Displacement" instead I get this:



Do I just need more of a gradient on my tile image? Or am I doing something more fundamentally wrong...

UPDATE:

Here's what the mesh looks like - it's not individual cubes.



Another Update
It really seems like a bug to me - if I can see the image mapped correctly as a diffuse color texture then why wouldn't it apply the corresponding bump map?

I've uploaded my packed .blend file here: https://www.dropbox.com/s/q3m23wurzeyqv19/rounded-grids-question.blend?dl=0
UPDATED ANSWER 1

Yeah, it might be a bug, you could report it into the bug tracker.

Here's the best workaround I could find:


set the influence to diffuse
create a second uv map (a quick Lightmap Pack should be fine)
bake into a uv-mapped texture (Bake mode: Textures)
now you can use this texture (in Cycles as well!) for the bumps


Cycles renders look better anyway, so you could consider this an improvement instead of a sad workaround :)

ANOTHER, EVEN BETTER, WAY TO DO IT

An hour later an even faster method occurred to me, where you don't need Blender Internal at all:


export-import as ply (the smaller faces in ply correspond to the original "grid")
add a second uv map. 
when you actually unwrap it, just select "Reset" - this assigns the full UV space to each face
use your original bump texture, which will be assigned to each grid face


ORIGINAL ANSWER 

The simplest way to do this is to add the Bevel Modifier to all of your cubes - you have a lot of control without messing with materials and textures. Increase the segments of the beveling until you are happy with the result.

The following question can help you to do this efficiently: Add the Same Modifier to Multiple Objects at Once?

If the OBJ format export-import creates triangles from the quads, then the bevel modifier could have problems. See this for converting back to quads: How do I convert a triangle mesh to a quads one?