When the summands of a positive definite matrix are positive definite

Question

Let $A,B$ be two real symmetric matrices. Let $C = A+B$ be a positive-definite matrix. Write $C>0$ for $C$ being positive-definite. Suppose that $A>0 \implies C>0$ and $B > 0 \implies C>0$, can anything non-trivial be said about when $$C > 0 \implies A>0? \hspace{1cm} \text{or} \hspace{1cm} C>0 \implies B>0?$$