Yes, it can mean the best, but it can also mean the thinist. An example of meaning the best would be, that's the finest example of art work I've ever seen.
An example of meaning thin would be, she has the finest hair out of all the women in the salon.
This can be confusing because the word fine is used as slang to mean good looking as well in American English. For example, she's the finest looking woman in the building.
I hope this helped đ