A $6$-dimensional vector space $\{ a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\}$
Let $V_1$ be $\{a_{1},a_{2},a_{3},a_{4}\}$
and $V_2$ be $\{a_{3},a_{4},a_{5},a_{6}\}$
$V_{1}\cap V_{2} = \{a_{3},a_{4}\}$
This is the smallest possible dimension, which is $2.$
The largest possible dimension will be $4$,when $V_{1} = V_{2}$