The problem of blind equalization was thought to have been solved by some `globally convergent' blind equalizers. However, the established local convergence of some blind equalizers demonstrates the possibility of ill-convergence in the true equalizer parameter space by these supposedly ideal algorithms. In this paper, we analyze the reason why an algorithm globally convergent in the combined channel and equalizer parameter space can have undesirable local convergence in the true equalizer parameter space. We demonstrate that the existing results on global convergence are implicitly based on an idealistic assumption that the equalizer is infinitely parameterized with infinite output delay. We show that despite the engineering intuition that such results should carry over to the finitely parametrized case; in reality the theory breaks down and local minima are common place. Thus, while the theories proving global convergence are elegant and positive, the reality is that for realizable systems such convergence behavior may not materialize.