Can we prepare alkyl chloride or alkyl bromide by Finkelstein reaction?
The thing is that you have to start with some compound $\ce{R-X}$ where $\ce{X}$ is a good leaving group. The quality of the leaving group is often thermodynamically defined, and in the Finkelstein reaction that you described, the energy motor for the whole thing is the insolubility of the salt that is produced.

I don't see what favourable starting material you could use for this in order to prepare other haloalkanes, especially since there are other ways for doing that. 

That being said, it would in principle be possible to do
$$\ce{R-X + NaY -&gt; R-Y + NaX v}$$
as long as the solubility of $\ce{NaY}$ is larger than the solubility of $\ce{NaX}$.