Current in a diode (LED or otherwise) obeys Shockley's diode equation (please check the link, I won't bother copying it here).
Parameters vary depending on the actual diode or LED, and also some "diodes" (as in: a physical component) actually contain several diodes in series, but the idea is that I varies exponentially with Vf.
This means we don't have a sharp, yes-or-no transition. Even at low Vf, some current will flow. It will be very small, but sometimes enough to produce a noticeable effect...
Let's do a quick simulation:

simulate this circuit – Schematic created using CircuitLab

I used a log axis for current, and a linear axis for voltage, which means the I-V characteristic is almost a straight line. At higher current, it is no longer straight, as the LED's internal resistance begins to matter (it is not modelled by Schockley's equation, but it is modeled by the simulator). The model used here was the default one in the simulator for a red LED, but it seems OK.
At 1V, we will still have 25µA current. With a modern high-brightness LED, and in low light, this will be visible to the naked eye.