  An interval $k$-graph is the intersection graph of a family $\mathcal{I}$ of
intervals of the real line partitioned into at most $k$ classes with vertices
adjacent if and only if their corresponding intervals intersect and belong to
different classes. In this paper we discuss the interval $k$-graphs that are
the incomparability graphs of orders; i.e., cocomparability interval $k$-graphs
or interval $k$-orders. Interval $2$-orders have been characterized in many
ways, but we show that analogous characterizations do not carry over to
interval $k$-orders, for $k > 2$. We describe the structure of interval
$k$-orders, for any $k$, characterize the interval $3$-orders (cocomparability
interval $3$-graphs) via one forbidden suborder (subgraph), and state a
conjecture for interval $k$-orders (any $k$) that would characterize them via
two forbidden suborders.
