I'm setting up my back speakers and I ran short of wire. I have extra wire but it is thicker than what I already ran. The speakers themselves already have the thinner wire built in. So, my question if you go from thin to a thicker gauge wire and back to thin do you get any distortion in the sound.

Not from changing gauge but from the splice you will need to make. Be sure you do as good a job as you can on the splice. I would solder if possible.

What Brian said. If you need to though, you can splice them using electrical tape as a temporary alternative.

Would butt connectors work I already have them from a radio I installed?