According to the California Department of Motor Vehicles, the minimum following distance on highways is three car lengths. Anything closer is considered tailgatinging and is both unsafe and against the law. This distance is independent of time of day.
If you can see the area lit by their headlights, then you are probably close enough to dim your own.
1000 ft
when you can see their headlights
Usually dim headlights are caused by a poor ground connection.
the courteous time to dim your headlights is as soon as you see another vehicle, or within one half mile.
when there are oncoming cars, it is appropriate to dim your headlights when they are about 300ft away. Check your diver manual for the most correct answer.
In most states it is 500 Feet. Common sense should prevail.
Your headlights may not dim because you don't have dimmer lights. Dimmer lights can blow and leave you with just the high beams.
You should dim your headlights within 500 feet of the car in front of you. The rule of thumb is, if you can see another car, dim your lights.
Here in California, by law it's 500 feet (a football field-and-a-half), but be kind and dim your bright lights well before that.
If by "dim headlights" you mean low-beams... the bulbs are bad and must be replaced.