Can I use a single 20-amp breaker for a bathroom and adjacent small bedroom in my basement?
Nope, you'll need 2 circuits for this

Bathroom receptacles must be on a 20A branch circuit that is either dedicated to bathroom receptacles (and no other loads), or to the loads in a single bathroom (and no other rooms), as per NEC 210.11(C)(3):


  (3) Bathroom Branch Circuits. In addition to the number of
  branch circuits required by other parts of this section, at least
  one 120-volt, 20-ampere branch circuit shall be provided to
  supply the bathroom(s) receptacle outlet(s). Such circuits shall
  have no other outlets.
  
  Exception: Where the 20-ampere circuit supplies a single bathroom,
  outlets for other equipment within the same bathroom shall be permitted
  to be supplied in accordance with 210.23(A)(1) and (A)(2).


As a result, the 20A circuit will need to feed the bathroom receptacles (and possibly the rest of the bathroom's loads as well), while all other loads (general receptacles outside the bathroom, and lighting loads) will go on the remaining 15A circuit.  All wiring on the 20A circuit will need to be 12AWG, as well.