Let $X$ be the underlying space of an irreducible scheme. In particular, $X$ is non-empty.
Note that the closure of any point is a closed irreducible subset. We say that a point has codimension $n$ if its closure has finite codimension equal to $n$. So, the unique point of codimension $0$ is the generic point.
Given a non-negative integer $n$, is it possible that there is a closed point in the closure of any point of codimension$\leq n$ but there is a point of codimension $n+1$ that has no closed points in its closure?