Does every vertex in a rigged model need to be weighted to a bone?

I have a mesh that has some static elements that aren't affected by any bones, but when imported into Unity, the static elements are attaching themselves to random bones.


Do I need to have the model re-rigged or is there an import setting that I'm missing?

I've tried setting the root node to random bones with no luck.

 

As an update, I had the models re-rigged to have a new bone that all the static vertices are weighted to. This appears to have fixed my problems.