Our paper investigates the issue of no-regret learning dynamics and their convergence. We show that convergence to a Nash Equilibrium that preserves the no-regret property is possible, but takes a long (exponential) time. In light of this result, we seek to extend typical Price of Anarchy bounds to the set of Coarse Correlated Equilibria (CCE) – which no-regret dynamics converge to – by comparing the cost of the best CCE to the optimum.