Well, vets are knowledgeable about medical topics for sure, but many aren't experts on dog breed identification, so their guess is likely as good as anybody's! Maybe they see a lot of border collie mixes in your area, and certainly many farm dogs have collie of some description in them, so it's a possibility. You'll probably never know for sure, but some of the big names in DNA testing for dogs are fairly reliable these days. My favorite is Embark, because you get genetic health screening results as well as an estimation of breed mix, and it also updates the results as their genetic database grows. Wisdom Panel is now offering a genetic health screening option with some of theirs, but I don't know much about it. Neither can tell you much about your dog's ancestry beyond three generations back, and they're certainly not 100% accurate yet, but they can be fun and informative if you want to have a better idea of what your pup's mix is.
If the farm has a Lab and a border collie, it is probable. However, "dog genetics" are a little different than people genetics. If the female dog were a Lab and mated with a BC, a Lab, and a Mix, the puppies could be the same as the mother, and various mixes with the other dogs, including Lab-Lab, Lab-BC, and Lab-Mix ... and so on.