For matrices of equal dimensions it holds that: $$rank(\textbf{A}+\textbf{B})\leq rank(\textbf{A})+rank(\textbf{B})$$
The equality generally holds when $C(\textbf{A})\cap C(\textbf{B})=\{\textbf{0}\}$ and $R(\textbf{A})\cap R(\textbf{B})=\{\textbf{0}\}$. Since this confuses me, I am trying to find some simple examples that the equality holds but I am stuck. I can only think of cases when one of $\textbf{A},\textbf{B}$ is zero.
Choose any numbers $a,b,c,d,e,f$ such that $af\ne b$ and $cd\ne e$. Consider $$\pmatrix{a&b\cr ac&bc\cr}+\pmatrix{d&df\cr e&ef\cr}=\pmatrix{a+d&b+df\cr ac+e&bc+ef\cr}$$ The two matrices on the left each have rank 1, their sum has rank 2 (since it has non-zero determinant).