Yes, there is a finite limit, but the number is
much higher than you would suppose.
First, in terms of spectrum, realistically all the satellites in a constellation can (and should) share a small number of relatively narrow frequencies. You can almost imagine it ma be the equivalent of one more cell tower in your area per constellation (at worse).
In terms of space, astronomy aside, the actual space these satellites are flying in are “huge”. As a perspective if you hold your thumb up to the sky, the distance “covered up” by your thumb (if I did my math right) is approximately 480 miles [1]
Yes things travel very quickly at those orbits, but the point is that even if we throw a ton of satellites up, they will remain fairly sparse.
That said, possible reflection and how noticeable they will be, that is a whole other question.
[1] assuming orbits of 366 miles up, higher orbits would be even farther apart