Would the length of the apothem ever be congruent to the length of your radius, and if so, when and why?

No, since the radius is just the radius of the circumscribed circle, the edge of the side would just create an arc on the circle, so its not the same length.

In other words, the length to a corner of the polygon will never equal the length to a side of the polygon.

Even a polygon with an infinite amount of sides, but still not a circle won't have a radius equal to the apothem, as there would be infinitesimally small arcs being formed on the circumscribed circle.