Bottlenecking won't ever reduce your performance after an upgrade. It might just mean that your performance won't increase as much as it could. If you have an X4 860K + GTX 950, upgrading to a GTX 1080 won't reduce performance. It'll probably help performance. But your CPU would be the bottleneck in your system, that is to say, it is the weakest link. It is holding your rig back and isn't pushing other components as fast as they can go.
To answer your question, the RX 480 is supposed to be a pretty powerful card. Will your CPU be a bottleneck? Probably. Will your system perform better than now? Also Probably. The most I've seen paired with an X4 860K is an R9 390. Looking at how the GFX market is shaping up (with significant performance gains on both AMD and Nvidia cards for similar price points), I'd probably look for a CPU upgrade before upgrading GPU.