  We prove that if a smooth variety with non-positive canonical class can be
embedded into a weighted projective space of dimension $n$ as a well formed
complete intersection and it is not an intersection with a linear cone therein,
then the weights of the weighted projective space do not exceed $n+1$. Based on
this bound we classify all smooth Fano complete intersections of dimensions $4$
and $5$, and compute their invariants.
