The Dushnik--Miller dimension of a poset $P$ is the least $d$ for which $P$ can be embedded into a product of $d$ chains. Lewis and Souza showed that the dimension of the divisibility order on the interval of integers $[N/κ, N]$ is bounded above by $κ(\logκ)^{1+o(1)}$ and below by $Ω((\logκ/\log\logκ)^2)$. We improve the upper bound to $O((\log κ)^3/(\log\logκ)^2).$ We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.