In Matrix Analysis by Horn & Johnson, the following condition is given: $range\left(A\right) \bigcap range\left(B\right) = \left\{0\right\}$ and $range\left(A^T\right) \bigcap range\left(B^T\right) = \left\{0\right\}$ I am trying to find a proof but no luck.
One direction is easy: $range\left(A+B\right) \subseteq range\left(A\right) + range\left(B\right)$ implies $rank\left(A+B\right) \le rank\left(A\right) + rank\left(B\right) - dim\left(range\left(A\right) \bigcap range\left(B\right)\right)$. If rank is additive, we have $range\left(A\right) \bigcap range\left(B\right) = \left\{0\right\}$. Taking transpose leads to $range\left(A^T\right) \bigcap range\left(B^T\right) = \left\{0\right\}$.