Sounds like it is the puller to me if all the teeth are still fine - the pullers can wear inside or bend out so they then can't guide the teeth together as closely as before. You would be able to tell for definite if you had another puller to compare it with - but try and have a look if the puller is painted black then the insides will have worn to bare metal, if excessively then that is probably the problem.
If it doesn't look that worn then it could be the top and bottom of the puller have bent away from each other - in which case a gentle tap with a hammer on a metal bolster could do the job - but it could also close the puller too much!
Pullers on zips often bend out when used on boots (specifically riding boots) as the tab you use to pull the puller up causes the main puller part to tilt (along with the awkward angle when doing up your own boots) which over time forces the two parts of the puller apart. Zips on coats etc there isn't this problem as the fabric bends before the puller can tilt.
The trick to keep zips from wearing is keep them clean and ensure the zip is held together before the puller is pulled so there isn't any stress on the puller which will wear the puller prematurely and could break it.
Got a bit technical there sorry! Lots of pulling going on!