BJT Mode of operation when VBE = VBE(ON)
For a BJT as you say if VBE &lt; VBE(ON) the transistor is in cut-off, when you get to VBE = VBE(ON) the base and the emiter of the transistor behave like a diode, with the voltage drop of a diode (0.6-0.7V) it means you "can't" have more than  VBE = VBE(ON), you only can have the voltage drop across the diode. If you try to increase the VBE you only increase the base current, and you can destroy the transistor.