can dropped voltage be greater than forward voltage?
Actually, Vf is rather arbitrary, and can be as low as you like - as long as you're talking about arbitrarily low currents. In general, for semiconductors diodes which are forward biased, 
$$I_f = I_0 e^{kV_f}$$
where $I_0$ is a scaling factor, and k depends on a bunch of things like material and temperature.

For "regular" diodes (silicon signal diodes) Vf is usually about .6 to .7 volts at a handy current level like 1 mA.

For LEDs, Vf depends on things like the LED material, and is usually specified at a nominal operating current like 10 or 20 mA. It is usually in the range of 1.5 volts (early red LEDs) to 3 to 3.6 volts (for 1 to 10 mA).

However, for greater currents the voltage drop does get greater.

So, your terminology is non-standard. Vf, in your terms, is not actually a threshold voltage, but rather a convenient way to characterize normal operation. The actual voltage during actual operation (Vd in your terms) can be either greater or less than the reference Vf, depending on what current level is used.