Yes: by definition, two adjacent angles have the same vertex, a side in common and don't overlap.
So, if two angles share the vertex and a side, but they overlap, they are not adjacent.
For example, consider the origin, the x and y axes, and the bisector of the first quadrant
Define angle between the x axis and the bisector, and angle between the x axis and the y axis. So, and have the same vertex (the origin) and one side in common (the x axis), but they do overlap, and are not adjacent.