In Wikipedia:
A subset $S$ of $\mathbb{R}^n$ is bounded with respect to the Euclidean distance if and only if it bounded as subset of $\mathbb{R}^n$ with the product order.
More generally, I was wondering for a set which is both a metric space and partially-ordered set, when the boundedness of a subset wrt the metric and wrt the order agree, or just one implies the other not the other way around?
Thanks and regards!
For a functional analyst, the natural setting for questions of this sort is the theory of Banach lattices, i.e., vector spaces with both a norm and a lattice structure. The typical examples are the $L^p$-spaces and these are very useful for looking at the kind of relationships between the two structures that seem to interest you. Particularly suggestive are the two extreme cases $p=1$ (where the norm is order continuous) and $p=\infty$ (where the property you mention---equivalence of boundedness in the topological and in the order sense---is valid).