It takes 20 seconds for the "Rested" buff to kick in. If it doesn't, then you've got some bug going on.

Nearby enemies will reset the rested buildup.

I have a vague memory of going to sleep once and waking up at dawn with an unlit fire. I can't remember if I got a rested bonus. A viking cannot go to sleep without a lit fire present. I am thinking that perhaps if one goes to sleep with just enough fuel to last for a few seconds then one may awake at dawn with no rested bonus.