The following is hearsay: A French woman once approached R' Joseph B. Soloveitchik (IIRCor maybe it was Rav Kook), ready to convert. After completing a rigorous course of study, she was prepared to accept the full burden of Jewish law and observance. R' Soloveitchik asked her if she was willing to embrace a Jewish identity and become a member of the Jewish People. She refused, proudly announcing: "I am French!" She then proceeded to sing La Marseillaise with gusto.
The conversion was denied.