Off the top of my head, I cannot think of a circumstance that would make this true. There isn't even 1 exception, especially if you are talking about an incircle and circumcircle for a triangle. They are constructed in entirely different ways which means that they can never be the same thing.
A circumcircle is constructed by using the 3 vertex points of the triangle. An incircle is constructed by using the angle bisectors and perpendiculars. Even with the most regular equiangular triangles are the centers coincident.
False.