Examples of this sort (and worse) are given, for dimension 5 and up, in http://scholar.google.com/scholar?cluster=14055290405510744870&hl=en&oi=scholarr
David Handelman [me], Effectiveness of an affine invariant for indecomposable integral polytopes, J Pure and Applied Algebra 66 (1990) 165–184, section 3, pp 16ff.
Specifically, this gives integral polytopes $K$ such that $e(K \cap Z^d) = eK \cap Z^d$ [the first denotes the sum of $e$ points in $K \cap Z^d$) for $e \leq d/2$, but for no $e > d/2$, and when $e \geq d/2$, $eK$ is projectively faithful (that is, its set of lattice points generates the standard copy of $Z^d$ as an abelian group) inside Euclidean space of dimension $d$. When the dimension is 6 or more, we can also assume $K$ itself is projectively faithful.
Towards the related question, any $M \geq d-1$ will do, as in my answer to Lattice points in dilated polytopes and sumsets.