I realised that the LED shouldn't be able to pass reverse current, but the capacitor shouldn't be able to pass DC.
So we have various possibilities.
1) If the LED does not conduct in reverse at all at the peak-peak voltage of the supply, the capacitor will charge up to the peak voltage, and then no more current will flow. That will not let the LED light, as no current will flow after the first few cycles while the capacitor is charging.
2) The capacitor is leaking some DC current. Then some current will flow and the LED will light, but the circuit is limited by the leakage current of the capacitor, so a big resistor would have the same effect.
3) The LED passes some reverse current.
4) Some combination of the 2) and 3)
Without taking measurements, I don't know exactly what is happening in your circuit, but you have to be getting reverse current in the LED or significant leakage in the capacitor.