A counterexample to the poset conjecture (or Neggers-Stanley conjecture) was found by Petter Brändén. Brändén's counterexample is not naturally labelled. On October 20, 2004, John Stembridge announced a counterexample to the naturally labelled case - some posets of width 2 (i.e., largest antichain of size 2) with 17 elements. Stembridge's paper is now available here (#52).