A graph algorithm is truly subquadratic if it runs in ${\cal O}(m^b)$ time on connected $m$-edge graphs, for some positive $b < 2$. Roditty and Vassilevska Williams (STOC'13) proved that under plausible complexity assumptions, there is no truly subquadratic algorithm for computing the diameter of general graphs. In this work, we present positive and negative results on the existence of such algorithms for computing the diameter on some special graph classes. Specifically, three vertices in a graph form an asteroidal triple (AT) if between any two of them there exists a path that avoids the closed neighbourhood of the third one. We call a graph AT-free if it does not contain an AT. We first prove that for all $m$-edge AT-free graphs, one can compute all the eccentricities in truly subquadratic ${\cal O}(m^{3/2})$ time. Then, we extend our study to several subclasses of chordal graphs -- all of them generalizing interval graphs in various ways --, as an attempt to understand which of the properties of AT-free graphs, or natural generalizations of the latter, can help in the design of fast algorithms for the diameter problem on broader graph classes. For instance, for all chordal graphs with a dominating shortest path, there is a linear-time algorithm for computing a diametral pair if the diameter is at least four. However, already for split graphs with a dominating edge, under plausible complexity assumptions, there is no truly subquadratic algorithm for deciding whether the diameter is either $2$ or $3$.