At what age do babies learn to "self-soothe?"?

My 6.5 month old daughter will not take a pacifier and if she's really upset, she has no interest in sucking her thumb. She used to suck her thumb to comfort herself a little bit (she knows how, at least), but now I wonder if teething is making that option not so comfortable for her. She's currently waking up 2-3 times at night (after a good 3-4 months of sleeping through or only waking once). I know that at this age she shouldn't be waking up 3 times to nurse, and I'd like to replace one of those late-night snacks with an alternate soothing thing. The thing is, if I nurse her while she's still drowsy, she falls asleep quickly. If I try other things (rubbing her back, giving her a soft toy to chew on), she just gets upset and then we all wake up and it's harder to get back to sleep. Any mommies out there on the other side of this phase?