DNA Database Completion Act of 2001 - Amends the DNA Analysis Backlog Elimination Act of 2000 to increase the authorization of appropriations for FY 2002 through 2004, and to authorize appropriations for FY 2005 and 2006, for grants to eligible States for DNA analyses of samples taken from individuals convicted of a qualifying State offense, and of samples from crime scenes, for inclusion in the Combined DNA Index System.