  A poset $P = (X,\prec)$ has an interval representation if each $x \in X$ can
be assigned a real interval $I_x$ so that $x \prec y$ in $P$ if and only if
$I_x$ lies completely to the left of $I_y$. Such orders are called
\emph{interval orders}. In this paper we give a surprisingly simple forbidden
poset characterization of those posets that have an interval representation in
which each interval length is either 0 or 1. In addition, for posets
$(X,\prec)$ with a weight of 1 or 2 assigned to each point, we characterize
those that have an interval representation in which for each $x \in X$ the
length of the interval assigned to $x$ equals the weight assigned to $x$. For
both these problems we can determine in polynomial time whether the desired
interval representation is possible and in the affirmative case, produce such a
representation.
