I have a 390x.
Simple answer: It won't be a bottleneck in the usual sense that the two components are mismatched and one is completely holding back the other (so my answer is no).
More detail:
Usually the GPU is always technically bottlenecking the CPU although sometimes its the other way around. So getting a more powerful GPU will in most games increase the frame rate, quality of the graphics and resolution you can play at.
At the moment I have pretty much ideal frame rates in all games at 1440p and at nearly always the highest quality settings. This will likely change in the next couple of years. I can see in rainbow six siege and witcher 3 that I would be better off (especially for siege) playing at 1080p.
If you are thinking of upgrading the 390x then I would recommend a card at the level of the 1080ti over a 1080. But you don't really need to upgrade especially if you are playing at 1080p and not too fussed at maxing out every setting. In pretty much all games you can change a load of settings and notice no difference but get a big boost in fps.
I wonder if you have considered the latest intel i7 or the 1600x? From the last benchmarks I saw the latest i7 is a better choice for gaming in the short term than the 1800x and the 1800x is comparable to the 1600x but at a higher price. In the future those 8 cores might matter in games but at the moment the 1600x performs more or less at the same level as the 1800x. Quite honestly I would not want to be buying a CPU at this time as I would be tempted to go the for the i7 as its better now but in 3 years who knows, optimisations for the AMD processors and moves to 6 or 8 cores in gaming might make the 1800x or 1600x a superior choice.