If you had a 24v 15mph 250w bike and put a 48v battery and controller on it, it'll try and go twice as fast to 30mph, but the motor can't hansle the power to go that sat, so it would get hot and burn.
Your basic idea about different voltages affecting speed is correct, but it also depends on the design speed of the motor. By winding the motor stator with different number of turns, the makers can get the motor to run at different speeds for the same voltage. The end result is that the motor will spin at so many rpm per volt. The user can then decide what speed it'll reach by choosing the appropriate voltage battery and controller. Obviously you need a lot of power to actually reach high speeds. Power = volts x amps, but as you know, if you push too many amps through a wire it gets hot unless its thick wire. Therefore, more powerful motors are generally bigger.