I'd almost say deer, but deer won't usually touch rosemary... too strong of a scent.

Some pi#%ed human?

There is a fence all around my yard and I always keep my gate closed so there is no way a deer could have gotten in. I used to get rabbits but I have since found and blocked the hole in the fence they were going through

Not deer, I have HERDS of deer and they never touch rosemary!

I've never seen anything like that on rosemary. I can't see a human vandal being bothered to pick off all those individual stems. How about some sort of bird? They do sometimes peck at plants for nesting material or in search of bugs. And they can be messy. Large things like magpies and pigeons can throw a lot of twigs about in search of the perfect one.

The broken pieces could be gathered up, washed and dried so at least they're not wasted.

I've watched starlings rip out my newly planted marigolds, and fly off with them. I'm guessing they love aromatic plants, because they either repel insects, or they like their nest to smell good.