Is it possible that the diameter of a graph be shorter than the longest shortest path?
The diameter of a graph is, by definition, the length of the longest shortest path, i.e.,

$$\mathrm{diam}(G) = \max_{x,y\in V(G)} \min\, \{\mathrm{len}(P)\mid P\text{ is a path from }x\text{ to }y\}\,.$$

But remember that the length of a path is the number of edges it contains, not the number of vertices. (That is, a single edge is a path of length&nbsp;1, not 2.)