Instinctively I'd say no. I found a counterexample, though it might seem a little contrived. See the attached pdf. The two 4x4 matrices both have rank 4, but their sum has rank 3.
It's true that if you generate a random nxn matrix, its rank will be n. You could generate random matrices in pairs, add them, take the rank of the sum, repeat this all day, you'd almost certainly find that every sum had a rank of n.
You seem to ask two questions, but it seems to me that the second question is just the first question in a different form.