Search online for an LED datasheet, and look for a current versus voltage curve. What you will find is that LEDs are very unlike ordinary resistive loads.
With an ordinary resistive load, the current flowing through the load increases pretty much linearly with the voltage applied to the load. With an LED, the current flow is pretty much zero, and stays zero, until you reach a threshold voltage, and then the current starts going up _very_ quickly. What this means is that around the proper drive voltage, the current changes dramatically. For example, if the proper drive voltage is 3.5V, then at 3.4V you might see only 1/4 the desired current, and at 3.6V the device would be severely overdriven.
The problem is that the proper drive voltage changes with numerous factors, including device production lot, age, temperature, heat sinking, etc. Your nominal 3.6V device might really need 3.5V or 3.7V, and the only thing that you can do is determine this once you have the device in hand.
There are really only two good ways to drive an LED. One is with some sort of current regulated power supply, which constantly adjusts the output voltage in order to maintain the proper drive current. The other is to place a resistor in series with the LED in order to make the current versus voltage characteristic less sensitive to small changes in LED voltage. Driving with a stiff voltage source is exactly wrong
One trick that is sometimes used is to drive the LED with a soft voltage source, meaning one in which the voltage drops as current increases...this is essentially a voltage source in series with a resistance (one of the basic techniques described above), but this resistance might (for example) be the internal resistance of a lithium cell.
-Jon