Some kinds of dimmers are hard to operate at low load (like an LED) and low settings. Your standard home-center lamp dimmer is like this. And some dimmer designers might choose not to go below a certain percentage.
PWM dimmers are not so limited. It should be easy for most PWM dimmers to go to 1% or less. I know some LED drivers claim as high as 3000:1 dimming ratios.
Most single-die or single-string LEDs can go to 1% or less on analog dimming. I've seen some go FAR below that.
However there are limits. It's well known that the spectral characteristics change as current changes. There's some debate about how significant this is with small changes in current, but there's little debate that as you get to extremes, it can be a substantial change. PWM dimming (done well) eliminates this issue.
In multi-string LEDs, e.g. a COB that has 10 strings of 10 dies, you may see the strings don't current balance well if you under-drive them substantially. This is more likely to be significant in "economy" LEDs than in name brands.
You may not care much about those, but here's a big one. Luminus specifies (or at least they used to) a minimum drive current that's more than 10% of the maximum drive current. I was informed that they could not guarantee that the LEDs would light correctly below that. I never completely bought into that, but it was pretty clear in their product literature, so pretty much a show-stopper for a professional design. This was important to me at the time because I was designing a driver for one of their LEDs that was intended to go below 10% on analog dimming. Maybe their lawyers made them say that because they saw one freaky die from a prototype batch a million years ago. Or maybe there is (was?) something more to it. I just don't know. That's the only manufacturer I've ever known to specify a minimum current for their LEDs.
With that said, there is (or at least can be) a minimum current below which an LED will suddenly stop emitting light. I have seen single strings of LEDs where some dies are emitting and others are not. In my case this occurred at EXTREMELY low currents - to the point where you had to dim the lights and get very close to tell which ones were on and which weren't. I suppose in deep darkness it would have been easy to tell, but I never tried that.
Lastly, in commercially available "LED bulbs" it's possible that they have internal drivers or similar circuitry that doesn't function well below some minimum dimming level.