  A nef-partition for a weighted complete intersection is a combinatorial
structure on its weights and degrees which is important for Mirror Symmetry. It
is known that nef-partitions exist for smooth well-formed Fano weighted
complete intersections of small dimension or codimension, and that in these
cases they are strong in the sense that they can be realized as fibers of
morphisms of weighted simplicial complexes, i.e., finite abstract simplicial
complexes equipped with a weight function.
  It was conjectured that this approach can be extended to the case of
arbitrary codimension. We show that in the case of any codimension greater than
3 strong nef-partitions may not exist, and provide a sufficient combinatorial
condition for existence of a strong nef-partition in terms of weights.
  We also show that the combinatorics of smooth well-formed weighted complete
intersections can be arbitrarily complicated from the point of view of
simplicial geometry.
