Let $X\subset\mathbb{P}^N$ be an irreducible projective variety. Let's denote by $\mathcal{T}$ the following property: through a general point $x\in X$ there is no line intersecting $X$ in at least three points counted with multiplicity (but not contained in $X$). I am assuming that $X$ is not a hypersurface.

Is there some natural condition on $X$ implying that $\mathcal{T}$ holds? For instance, if $X$ is linearly normal does then $\mathcal{T}$ necessarily hold? Is there a classification of the varieties $X$ for which $\mathcal{T}$ does not hold?