Well, depends on what resolution and what you will be doing.
But if you are thinking of 1080p the answer is definitive YES, it will bottleneck.
However most games will have no problem running on 60fps+ and there won't be any issue same is If you will be doing things in 1440p and hoping for stable 60fps this simply won't be a problem.
To some degree 4K video while not great, will be doable.
However if you will be chasing high fps for latest titles on 1080p you will be disappointed. While in games like CS:GO you should be fine, if you are trying to hit 144fps (for that new 1080p curved monitor it just not going to happen).
That said for video editing and 3D rendering, you will be fine its just you won't be getting all you paid for.
With that said if you are thinking of upgrade and you already have the gtx 1070, just do it. Put it in, it won't hurt you in fact it will allow you to increase the game details, just not the FPS.
But honestly if you got the system for dirty cheap I would recommend you to do is to clean the HDD and sell the whole thing as it is pretty decent machine to sell for decent money given the gtx 1060 in it.
And get yourself a R5 1600,2600 pair it with the 1070.