When a person says "you can't teach an old dog new tricks," he or she means that some one will have great difficulty in learning some specific thing (and thus should not waste the time trying to learn the thing), for most old dogs do indeed have trouble learning new things. There are of course, exceptions, such as my dog, Pierre. Pierre came to my household at the young age of nine months. At that time we were keen on training him, since "you can't teach an old dog new tricks." We proceeded to teach the dog how to sit, and how to go up on his hind legs. However, we had trouble teaching him how to heel. We got lazy and would go out with him without making the effort of forcing him to heel. Soon we began justifying our laziness by saying that it was too late--saying that the dog was now accustomed to not heeling and that you can't teach an old dog new tricks. Pierre, now ten years old, still does not know how to heel. He has however, shown me that you can teach an old dog new tricks. First of all, a few years after we got the dog, my family decided to remodel the house. With the remodeling, we installed off-white carpeting into the bedrooms. My mother declared that the dog would not be allowed to walk into the bedrooms, to keep the carpeting clean. Either he had to learn, or he had to go. At first we had to carefully close all the bedroom doors. But with time Pierre knew that he was not supposed to walk on the carpeting. We could throw his bone onto the carpet and he would run towards it until he hit the edge of the carpet, at which point he would stop and just look pitiful (pitifully cute), wantig someone to get his bone for him. I do not mean to imply that the dog never walks on the carpet. Indeed, he does if no one is home and he wants to go get a shoe (he has a shoe fetish). But the point is that he is well aware that he is not allowed to walk on the carpet. When he walks on the carpet, he is conscious that he is trespassing, but sometimes a dog has to do what a dog has to do.
One could claim that my carpet example does not violate the adage, for Pierre was not really old at the time he learned to stay off the near white carpet. Yet today he is really old, and he still keeps learning. Right now I am in the middle of teaching him how to walk around Boston without a leash, without being hit by cars. It is not exactlly a last attempt at teaching him how to heel, for I let him stop and run ahead of me as much as he wants. All that I am teaching him now is to stop at roads, and wait for me before he attempts to cross them. It is going well. Someties he is oblivious and I have to sharply call out to him to make him stop, but at least he stops half the time. A few months ago he had no concept of roads versus sidewalks. So indeed, you can teach an old dog new tricks. Of course Pierre is somewhat of an exception. First of all, he is a mutt, and mutts are just far more intelligent and superior to inbred, problematic pure breed dogs. Second of all, he grew up in an exemplary household. So perhaps Fifi will never learn new tricks, but with Pierre it is a different story.