Special secants to curves

Let $$X\subset\mathbb{P}^n$$ be a smooth nondegenerate (i.e. not contained in any hyperplane) curve over $$\mathbb{C}$$. Is it possible that every collection of $$n-3$$ points on $$X$$ lies on a $$n-1$$-secant (i.e. a linear space spanned by $$n-1$$ points on $$X$$) that has dimension at most $$n-3$$? If yes, can $$X$$ even be chosen to be projectively normal?

For example, for $$n=4$$, is there a curve $$X\subset\mathbb{P}^4$$ such that every $$p\in X$$ lies on a trisecant line?

My hope and intuition is that this cannot happen but I don't have a proof.

I will give an example for $$n = 4$$. Let $$S = \mathbb{P}_{\mathbb{P}^1}(\mathcal{O}(-1) \oplus \mathcal{O}(-2)) \subset \mathbb{P}^4$$ be a cubic scroll and let $$X \subset S$$ be a curve which is a 3-section of the morphism $$S \to \mathbb{P}^1$$. Then every point of $$X$$ lies on a trisecant line (the ruling of the scroll).