OK, we get back to the matter - frequently discussed here - of what a LED is.
You should perhaps research it more thoroughly, but in essence a LED is a diode with a threshold before which it will not conduct at all and at which it will readily conduct large currents with minimal increase in voltage above that threshold, thus having a very low effective dynamic (series) resistance.
Your situation therefore is that the threshold voltage of the LED is very close to the voltage available from your regulator. This is a completely impractical way to drive a LED as depending on circumstances including the particular parameters of that LED and the voltage regulator, and the temperature, the LED might variously draw very little current as you describe, or possibly a heavy current.
In general, you need to control the current by allowing something of the order of a volt across the dropping resistor, so for operating 20 mA LEDs, you want 4.5 or 5 V. In your case, you have successfully used the internal 40 Ohm (at 3.3 V) dynamic resistance of the ATmega for current limiting, but this situation is extremely sensitive to the supply voltage and could easily exceed the rating of the ATmega with a relatively small increase in supply voltage.
That said, it is somewhat fortunate that the ATmega has a positive temperature coefficient - as it heats up, its ESR increases and limits the current more.
And finally, 13 mA is plenty to operate a LED; doubling the current would produce a noticeable, but not stunning increase in brightness.