Chains intersecting antichains in finite posets

I feel a little embarrassed to be asking this question here, since I think it should be much easier than I'm making it, but here goes:

Given a finite poset P, does there necessarily exist some chain that intersects every maximal antichain? (Note: By maximal antichain, I mean that there's no antichain strictly containing our antichain.) The answer seems to be "no" for infinite posets, but I can't find either a reference or a proof when it comes to finite posets.

Sorry if this is an undergrad-homework-level problem...

-
On the other hand, there does exists a chain that intersects every antichain of maximum size. –  Richard Stanley Feb 5 '11 at 4:00