Latest info: I have now obtained some plain, no wiring, no resistor, 5mm white LEDs. The pack is marked 3.2V – 3.8v.
Applying an increasing voltage gives light output at 1.8v, dim, at which point the milliammeter registers. Increasing to 3.2v and the brightness becomes 'standard – not measured but what I have come to expect visually – and the current rises to 17mA. Increase again and the brightness does improve but not a lot, and the current at 3,8v is 27mA.
Increase the voltage further and at around 5v the light output starts to dim, losing the whiteness and becoming bluish. Then after about 20 seconds at about 6.5v the light stops and the current rises to over 100mA.
Turn off for 5 minutes. Turn on again – current at 3.2V is now 69mA, and at 3.8v is 84mA. These currents seem to stay steady – and I conclude that failure involves the destruction of the phosphor coating, and causes something else inside to change reducing the 'resistance'. But a failed LED does not go open-circuit.
I hope this is useful to those among us who are not yet up to speed on 'modern' wizardry.
Cheers, Tim