Can a pair of angles be both vertical and complementary at the same time?

No, a pair of angles cannot be both vertical and complementary at the same time. Vertical angles are formed by the intersection of two lines and are always equal in measure. Complementary angles, on the other hand, are two angles whose measures add up to 90 degrees. Since vertical angles are equal, they would only be complementary if each angle measures 45 degrees, which is not generally the case.