Is a directed set countable, if for each element there are only finitely many smaller ones?
Solution 1:
Let $X$ be any set, and let $A$ be the collection of finite subsets of $X$; $A$ is directed by $\subseteq$, and each member of $A$ has only finitely many predecessors in that order. However, if $X$ is infinite, then $|A|=|X|$, so the cardinality of $A$ can be as large as you like.