  For each $n,r \geq 0$, let $KG(n,r)$ denote the Kneser Graph; that whose
vertices are labeled by $r$-element subsets of $n$, and whose edges indicate
that the corresponding subsets are disjoint. Fixing $r$ and allowing $n$ to
vary, one obtains a family of nested graphs, each equipped with a natural
action by a symmetric group $\mathfrak{S}_n$, such that these actions are
compatible and transitive. Families of graphs of this form were introduced by
the authors in [RW], while a systematic study of random walks on these families
were considered in [RW2]. In this paper we illustrate that these random walks
never exhibit the so-called product condition, and therefore also never display
total variation cutoff as defined by Aldous and Diaconis [AD]. In particular,
we provide a large family of algebro-combinatorially motivated examples of
collections of Markov chains which satisfy some well-known algebraic heuristics
for cutoff, while not actually having the property.
