  When can we compute the diameter of a graph in quasi linear time? We address
this question for the class of {\em split graphs}, that we observe to be the
hardest instances for deciding whether the diameter is at most two. We stress
that although the diameter of a non-complete split graph can only be either $2$
or $3$, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute
the diameter of an $n$-vertex $m$-edge split graph in less than quadratic time
-- in the size $n+m$ of the input. Therefore it is worth to study the
complexity of diameter computation on {\em subclasses} of split graphs, in
order to better understand the complexity border. Specifically, we consider the
split graphs with bounded {\em clique-interval number} and their complements,
with the former being a natural variation of the concept of interval number for
split graphs that we introduce in this paper. We first discuss the relations
between the clique-interval number and other graph invariants such as the
classic interval number of graphs, the treewidth, the {\em VC-dimension} and
the {\em stabbing number} of a related hypergraph. Then, in part based on these
above relations, we almost completely settle the complexity of diameter
computation on these subclasses of split graphs: - For the $k$-clique-interval
split graphs, we can compute their diameter in truly subquadratic time if
$k={\cal O}(1)$, and even in quasi linear time if $k=o(\log{n})$ and in
addition a corresponding ordering of the vertices in the clique is given.
However, under SETH this cannot be done in truly subquadratic time for any $k =
\omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split
graphs, we can compute their diameter in truly subquadratic time if $k={\cal
O}(1)$, and even in time ${\cal O}(km)$ if a corresponding ordering of the
vertices in the stable set is given. Again this latter result is optimal under
SETH up to polylogarithmic factors. Our findings raise the question whether a
$k$-clique interval ordering can always be computed in quasi linear time. We
prove that it is the case for $k=1$ and for some subclasses such as
bounded-treewidth split graphs, threshold graphs and comparability split
graphs. Finally, we prove that some important subclasses of split graphs --
including the ones mentioned above -- have a bounded clique-interval number.
