All dogs can be as great as Lassie! Photo credit: Fran Jewell

Absolutely! Although I refuse to tell anyone that their dog cannot be trained.
What makes a dog untrainable? Free access to furniture, sleeping areas that belong to others, food, height, affection, and any resources the dog has claimed as his own. Notice the key words here are FREE ACCESS. When a dog has free access anytime to the things that are important to him, then he will no longer have any respect for his owner or others, including people and other dogs. He will have no interest in learning or working.
There is a simple term for this. It’s called “spoiled.” Spoiled dogs have no need to work, to learn or do anything because everything important to them comes free.
Dogs are opportunists. They are fully capable of determining what is easiest to get and what they want. So, if there is nothing valuable to “work for,” then it is impossible to train them.
It is my belief that when someone comes to me with a difficult dog to train, a huge portion of the training becomes “un-spoiling” them or making them work for anything that is important to that dog. This is called “LEADERSHIP.”
It is further my belief that almost every problem a dog has stems from lack of leadership IN A MANNER THAT THE DOG UNDERSTANDS.
In very, very unusual cases, there are dogs that are so fearful they cannot get through the fear to trust someone to make learning possible. Studies are now showing that fearful temperament is mostly genetic. Environment does play a part, but not as much as genetics do. There are dogs that have been extremely abused and come out smelling like a rose as if it never happened. Then there are dogs that act as if they were abused, but were not, and can only process slight learning outside their fear.
And then there are cases of dogs that have grown up feral that never learned HOW to learn. These dogs usually spend all their time operating from a place of total instinct because it is the only way they know how to survive.
So, are there dogs that are untrainable? I still contend that all dogs can be trained and learn, but ONLY if the owner is willing to NOT spoil the dog in the first place, or “unspoil” the dog that is already spoiled. Sadly, if the owner is not willing to do what is necessary to train the dog, then yes, the dog will most likely be untrainable.