Let $S=\{A_{1}, ..., A_{m}\}$ be a set of $n \times n$ symmetric matrices and $m>n$, the rank $r(A_{i})=1$ for each $i$. Suppose that for any $m-1$ matrices $\{A_{i_{1}},...,A_{i_{m-1}}\}$ in $S$, we have the rank of matrix $r(\Sigma_{j}A_{i_{j}}) \leq t$.

My question is:

$\Sigma_{i} r(A_{i}) \leq t$ ?

I think this is true for $n \leq 2$, but I have no idea to prove the general case.