Yes, indeed. (Even the cats we have had were rescues.)
A year after we were married my wife and I adopted as a companion for our Golden a wonderful 10-year old girl, Rae-Rae, who had been living in the rescue kennel for almost a year because she was old, she had unsightly (though benign) tumors on her teats, and she had horrible teeth. But she had beautiful deep brown eyes that sparkled and laughed at us and she delighted in our attention. She was a true angel
. We had her for three years before a cancer took her from us suddenly, less than a month after we lost our golden retriever.
Since then we have taken in mostly tough sells - one big girl who was terrified of the wind and the feel of grass under her feet, another girl who was dropped off after she had damaged her back from a fall and couldn't walk properly. One girl we adopted, also 10, had developed bone cancer; she was with us for almost two months and had just started feeling happy and excited when her humans came home when things came to a head. Johnson had developed seizures, and Athena was 9 when her owner became tired of her.
Our times with our dogs are short, but that's just a reason to treasure every moment. I think that lesson has helped us with our kids, who we adopted in our late 40s; there are often times when I have to grit my teeth and tell myself "treasure this moment!"