Given two points $x,y$ in a connected scheme $X$, does there exist a finite sequence of affine opens which "connect $x$ to $y$"?

Question

Let $X$ be a connected scheme, and let $x,y\in X$ be two points.

Can we find a finite set of open affines $U_0,\ldots,U_n\subset X$ with the properties:

1. $U_i\cap U_{i+1}\ne\emptyset$ for $i = 0,\ldots,n-1$
2. $x\in U_0$, and $y\in U_n$.

I don't want to make any Noetherian assumptions.