Seeing "too much noise" is partly a matter of taste, it's very subjective. One person may think the noise at 1600 ISO is fine and another may not like it at all. And how large the image will be seen affects how distracting noise will be, and how much the blacks or overall image, or certain colors, will be lifted and/or lowered will affect noise. The higher the ISO, the less latitude you have to make changes in post to levels before the noise gets visibly increased, in other words, you may find it usable at 1600 ISO in tests only to find that you slightly underexposed the scene or didn't use enough fill light and once you make the correction in post, the noise goes up because you are more or less rating it even higher in one or more of the color channels.
On the other hand, you can apply some noise reduction in post-processing, though it may impact sharpness.
800 ISO is a good starting point for low-light work, I use that all the time on the Alexa and occasionally switch to 1000 ISO without that jumping out too much, but lower ISO's are definitely cleaner. If you want a very clean look, then 400 or 500 ISO is good.