Some Rabbis without any jewish roots; all of them are converts to judaism (gerrim):
Rabbi Dr. Asher Wade (Chassidut Gur/Ger)
http://www.youtube.com/watch?v=nszEJE70dvk
Rabbi Natan Gamedze (African black haredi litvish Rabbi)
http://www.youtube.com/watch?v=7qtid87l4xY
Rabbi Yaakov Ephraim Parisi (Chassidut Chabad-Lubavitch)
http://www.chabad.org/multimedia/media_cdo/aid/1416870/jewish/A-Pastors-Journey-to-Judaism.htm
Rabbi Moshe Hattori (japanese convert to orthodox Judaism)
http://www.youtube.com/watch?v=RPLsmwEbEjk
Rabbi Binyamin Klugger (french convert and Chabad-Lubavitch Rabbi)
Rabbi YY Rubinstein
So gerrim CAN become rabbis!