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 Kadison-Singer problem. See
and the references therein.
Your second question is Problem 2.2 there.
The questions themselves are not obvious ones. Why did you ask them?