Most typical LEDs are rated for 20 mA max, that doesn't mean they will pop at 21mA. It just means that once you exceed that rating the lifespan of the component is reduced. It won't "pop" until you greatly exceed that rating, but it will certainly be damaged.

I'm not sure I understand your method for measuring voltage drop. A typical white LED should be something in the range of 3.5v. When lit, you should be able to read roughly a 3.5v difference between the anode and cathode. If your fully-charged battery source is at 6.4v then that means you need to drop the remaining 2.9v with the resistor. R = V / I, so the desired resistance is 2.9v/0.02A = 145 ohms. Anything less than that and you risk overdriving the LED. And even though the current rating is usually 20mA, most are efficient enough to where you almost can't tell the difference between ~15mA and ~20mA.