Is every unit norm Bessel sequence in a Hilbert space a finite union of separated ones? Is every unit norm separated sequence a finite union of uniformly minimal (minimal with uniformly bounded biorthogonal vectors) ones?

Your questions are weakenings of the Feichtinger conjecture, which is equivalent to the KadisonSinger problem. See http://www.aimath.org/WWN/kadisonsinger/FrameProblems.pdf and the references therein. Your second question is Problem 2.2 there. The questions themselves are not obvious ones. Why did you ask them? 

