Wikipedia's article about Radon's theorem and its related states:
Carathéodory's theorem states that any point in the convex hull of some set of points is also within the convex hull of a subset of at most d + 1 of the points; that is, that the given point is part of a Radon partition in which it is a singleton.
But what about four points in the plane arranged like the vertices of a square? Each one belongs to the convex hull of the four points. But there is no way to take one point and have a Radon partition, isn't it? Am I wrong, or is the article wrong?
You are right, and the part following "that is," is misleading. Your statement about the vertices of a square is correct.
The intended meaning surely is the following: For any set $S \subset \mathbb{R}^d$ with convex hull $C$ and any point $p \in C \setminus S$ there are $d+1$ points $\{x_1,\dots,x_{d+1}\}$ from $S$ such that $\{p\}$ and $\{x_1,\dots,x_{d+1}\}$ form a Radon partition of $\{p,x_1,\dots,x_{d+1}\}$.