Actually, the property is even stronger than that: if X is a connected space and C is a nontrivial connected subset of X, C has a point that is a cut point not only of C, but of every connected subset of X containing C.

Here's how I ran across the problem above: as is well known, if X is a compact Hausdorff space and a, b are in separate components of X, then a and b are in separate quasicomponents of X.  In my case, start with a continuum X (= connected + compact + Hausdorff) with the property that whenever a and b are distinct, there is a third point c such that a and b lie in separate components of X-{c}.  Can we always conclude that a and b lie in separate quasicomponents of X-{c}?  The answer is YES.