The reason why you have not been able to get a definite answer to your question is if you don't know what the specifications for your LED are, then you are forced to make some assumptions. This is why manufacturers issue data sheets on their products. But LEDs, unlike many other components, are difficult to mark with their IDs. A resistor can have colored bands or numbers. A transistor can have a part number. An IC can have a part number. A capacitor can have a value and Voltage rating.
But most LEDs do not have any markings and once they are separated from a bag or other container with the part number, they are just a mystery part. Operating current? Unknown! Light level output? Unknown! Forward Voltage? Exact value is unknown! If you have enough of them you can run your own tests to the point of destruction. But if you have only one or a very few, then you are forced to just GUESS. Unfortunately, LEDs are usually so small that it is very difficult to impossible to observe things like the die temperature as they are operated at increasing current levels. So total failure is usually the first indication that you will have when testing them.
So let me guess. You have purchased a bargain bag of assorted LEDs and want to use them. I have done the same thing. And I just GUESS. Most LEDs will work with a current of around 15 or 20 mA. So that is where I would start. I set my lab power supply for it's minimum current (0 - 5 mA) and the Voltage to 5 to 10 Volts. Most LEDs will have a forward Voltage under 5 Volts so that should be high enough to allow them to draw the current as set by the supply's limit. I apply this limited current power and observe the LED. If it is bright, I assume that current is where it should be operated and stop. If it is dim I crank up the current limit until it appears to be bright enough. How bright is that? Well, you have seen other LEDs, haven't you. It is a GUESS!
If you have several of the same LED, you can crank up the current, carefully watching the current meter, until the first one blows. Then turn the current control down and hook up a second one. Crank it up to 1/2 the current where the first one released the smoke and leave it on overnight. If it is still OK in the morning, that current is probably OK. If it also burned out, then try again at 1/4 the original current. Repeat as needed.
Other than destructive testing, there is no way to determine the exact characteristics of an unknown LED.
SBD! (Sorry Bout Dat!)
And if you want to work with definite specifications then you need to buy known parts with available specification sheets. That's just the way things are. And brand, spanking, NEW LEDs with known part numbers are DIRT CHEAP! Well, usually, but don't ask.