Linear section of an algebraic variety

Question

Let $\pi$ be a linear subspace of $\mathbb{P}^n$ and $X$ a reduced, irreducible variety of $\mathbb{P}^n$. Suppose that $\pi\cap X$ is reducible, hence $\pi\cap X=Y_1∪Y_2∪⋯Y_k$. When the degree of $Y_i$ is "small", i.e., when $d<\dim\pi−\dim Y_i+1$, then we have that $⟨Y_i⟩⊊π$. My question is: can we have $⟨Y_i⟩⊊π$ if $d≥\dim π−\dim Y_i+1$?