I am testing a board powered by 12V and 5V supplies. I see the voltage drop across my led is 12V. Can this happen?
Unless your LED is one of those big multi-watt jobbers, then no, it cannot happen (assuming the LED is just an LED). Somewhere on your board is a series resistor which is dropping about 9-10 volts, leaving 2-3 volts on the LED itself. There are some LEDs which have an internal IC which regulates current. These used to be available 30 years ago, but I don't know if they're still around. I suspect they went out of style due to heating of the LED.
If the LED is a panel mount (rather than PC mount) then it's entirely posssible that there is a resistor built in, in which case 12 volts is applied to the unit, but only 2 or 3 volts per LED inside.
Couple of possible options here. One is that the LED contains a current limiting resistor or is built into an assembly that contains a current limiting resistor, and you're measuring across the LED and resistor in series. Another option is that the LED has failed open and no current is flowing, hence you will measure the supply voltage across the LED as the resistor is not dropping any voltage. Finally, you may be connecting the probes incorrectly and you aren't measuring what you think you're measuring.