In this paper, we study the evolution over time of the correlation structure of equity returns by means of a filtered-network approach and use this to investigate persistency and recurrences and their implications for risk-diversification strategies. We build dynamically planar maximally filtered graphs from the correlation structure over a rolling window and study the persistence of the associated directed bubble hierarchical tree (DBHT) clustering structure.We observe that the DBHT clustering structure is quite stable during the early 2000s, becoming gradually less persistent before the unfolding of the 2007-8 crisis. The correlation structure eventually recovers persistence in the aftermath of the crisis, setting up a new phase, distinct from the precrisis structure, in which the market structure is less related to industrial sector activity. We observe that the correlation structure is again losing persistence, which indicates the building up of another, different phase. Such dynamical changes in persistency and its occurrence at the unfolding of financial crises raise concerns about the effectiveness of correlation-based portfolio management tools for risk diversification.