By computing non-vanishing dual Stiefel-Whitney classes of the incomplete real flag manifold of length 3, $\mathbb{R} F(1,1,1,n-3)$, $n>4$, we obtain non-immersion and non-embedding results for the manifold and give solution to the immersion / embedding problem for $n=5, 6$ and $7$ by showing that Lam's estimate are best possible for these.