They can be constructed. (It's no use defining a family counterexamples that are impossible to construct; otherwise they can't be counterexamples!) The simplest such polygon consists of four points (p_{1}, p_{2}, p_{3}, p_{4}) with p_{2} near one intersection and p_{4} near the other interesection of the circles centered at p_{1} and p_{3} each of radius d (p_{1}, p_{3}). See Figure 6. Thus, p_{1} and p_{3} are our p_{i} and p_{j} and p_{2} and p_{4} alone achieve the diameter which is slightly less than d (p_{1}, p_{3}).
Because of the way the polygon is constructed, the point furthest from p_{i} is p_{j}, and vice versa. Recall that all points other than p_{i} and p_{j} lie inside the circles centered at p_{i} and p_{j} of radius d (p_{i}, p_{j}). As a result, if the algorithm begins at A = p_{i} then it will find B = p_{j} at a maximum distance from A = p_{i}. Then, when it searches for a point at a maximum distance from B = p_{j} it finds p_{i} so the algorithm incorrectly terminates with A = p_{i} and B = p_{j} as two points achieving the diameter.
See if you can construct one of the polygons described above
and then watch how the Snyder and Tang algorithm handles it.