Ptolemaic distances are distances for which the property
$$ \overline{x_1x_3} \cdot \overline{x_2x_4} \leq \overline{x_1x_2} \cdot \overline{x_3x_4} + \overline{x_2x_3} \cdot \overline{x_1x_4} $$
holds for any four points $x_1, x_2, x_3, x_4$. Examples include the euclidean metric, more generally all metrics induced by an inner product. Metrics which are not ptolemaic include the $L_p$-distances for $p \neq 2$. In fact, the $L_p$-distances are neither metric nor ptolemaic for $p \in (0, 1)$.
This leaves but a single category: Distances which are ptolemaic but not metric. Are there any examples of this?
Perhaps a rather silly example, but there is a non-metric ptolemaic distance on a set $\{x_1,x_2,x_3,x_4\}$ with only four elements.
Let the distance from $x_1$ to $x_2$ be $1000000$, the distance from $x_3$ to $x_4$ be $0.000001$ and all other distances between distinct points be $1$.