At my home club in Charlotte, Carolina GC, golfers are asked to fill divots in the Bermuda with a sand mix. Overfilling happens sometimes, and that's not good. I've seen divots were some golfers have tried to replace the divot, and it doesn't seem to work. Of course, I suppose you wouldn't notice where it did work.
I play another [cooler climate, not in Charlotte] course a lot that has bentgrass fairways. There they ask you to replace divots in the fairways. Some seem to take hold and some don't. They also give you a mix to use on the tees -- only. I don't know what kind of grass is on the tees, but it is definitely not the same as on the fairways.