As far as I can tell, the idea to use filtered categories instead of just ordinal-indexed diagrams is due to Grothendieck. In fact, let me use this opportunity to advertise my favorite text on abstract category theory: Exposé I of SGA 4. It rocks.
Anyway: Proposition 8.1.6 (loc. cit.), which Grothendieck attributes to Deligne, says that every filtered category receives a cofinal map from an ordered set.
Grothendieck remarks that, while this result says that the two points of view on filtered objects (general or ordered) are essentially equivalent, filtered objects arise more naturally.
A nice example is Grothendieck's Theorem 8.3.3 on ind-representability. Aside from some set-theoretic issues, this basically boils down to the statement that exactness of a presheaf of sets $F$ on a category $C$ is equivalent to the category $C_{/F}$ being filtered.