Does a led determine itself current load?
The relationship between voltage and current in a LED looks like an exponential curve. The picture below shows the relationship for a LED with a 2.0V forward voltage @ 20mA.



If this was connected to a 2.5V power source with no current limiting resistor, the amount of current flowing would be very high, much higher than the LED's rated peak forward current. The LED would fail catastrophically, probably with a pop and some smoke.*

If the the power supply was 2.5V @ say 10mA (a very small current that you probably wouldn't find) then the LED would draw more current at 2.5V than could be supplied, and the output voltage of the supply would drop.

*When I was a kid I had a 200-in-1 electronics project kit. As the batteries wore down I kept reducing the LED resistors until eventually I had no resistor at all. Then one day I changed the batteries and every LED blew. I was devastated.