This statement is slightly ambiguous, and wrong or right depending on how you interpret it. If you have an object M (such like the character mesh in your picture), and the minimal enclosing OBB O for the object M, then the following statements hold:
So you are saying that with any minimum OBB, taking the length of the vector formed by the 3 extents will give a minimum bounding sphere?
1) The sphere S which minimally encloses O is computed by taking the center of O as the center of sphere, and using as the radius half the length of the diagonal of O.
2) Even if S computed like above minimally encloses O, it most often is not the minimal enclosing sphere for the object M. (It can be quite bad, as you noticed)
So you see that the "minimally enclosing" property is not transitive. If S minimally encloses O, and O minimally encloses M, then it does not follow that S also minimally encloses M.
To produce a minimal enclosing sphere, see for example the Miniball code, or Welzl's algorithm. For a fast and very good approximation, see the Ritter's algorithm.