Let $V_1, V_2, ..., V_n$ be $t$-dimensional sub-spaces of an $n$-dimensional vector space $V$ where $t \lt n$.

Under what conditions the following would be true:

for any $B= \{v_1, v_2, ..., v_n\}$ a basis of $V$, there is a permutation $\pi \in S_n$ such that

for all $1 \leq i \leq n$, dim$($span$(V_i \cup v_{\pi(i)})) = t +1$ ?