It’s a degrees of freedom argument. If you don’t have as many degrees of freedom in your information as you do in the object you’re describing, then you can be sure that you haven’t completely specified the object. If you have more degrees of freedom in your info than your object, then you probably have specified it. And usually, if you have the same number of degrees of freedom in both, you’ve specified your object up to a finite number of discreet possibilities.
But k=3*n is off. The object we’re discussing has arbtrary position and orientation, so the coordinates of the first ball you place are completely arbitrary, and the second and third balls are somewhat arbitrary. Really, a “complete” (to our standards) specification of the position of all balls only has 3n - 6 degrees of freedom.
So, for four balls, we have six edge lengths and six independent ball coordinates, so we have a few discreet possible ball configurations (I think there are 6!/24, or 30, of them), but with seven edge lengths, we probably have it nailed down. For any given number of edge lengths, there will always be special cases where we don’t have enough information, but those special cases are of measure zero.
As an example of such a pathological cases, suppose that n-3 of our n balls have the exact same position (and therefore, (n-3)(n-4)/2 of our n(n-1)/2 edges are of length zero). This problem is equivalent to the 4 ball problem, which we already showed we couldn’t solve.