So the LTC3105 [1] looks to be able to supply 0.060 amps at 4 volts. Bluetooth LE [2] seems to use about 0.015 amps at peak (voltage not listed). If I am reading this correctly, it is possible. The question is: what actually runs the circuit connected to the Bluetooth transceiver?
[1] http://gnodevel.ugent.be/crr.ugent.be/archives/1628
[2] http://en.wikipedia.org/wiki/Bluetooth_low_energy#Technical_...