For a large collection of small examples, let $D$ be any $2\times 2$ matrix that has an inverse (Theorem TTMI can help you construct such a matrix, $I_2$ is a simple choice). Set $A=D$ and $B=(-1)D$. While $\inverse{A}$ and $\inverse{B}$ both exist, what is $\inverse{\left(A+B\right)}$?
For a large collection of examples of any size, consider $A=B=I_n$. Can the proposed statement be salvaged to become a theorem?