I think a lot depends on the individual dog. I had surgery last month and Faith and Mac were all clingy and snuggly like "Mom what's wrong?" Slider and his evil spawn, Bruiser, were like "Hey Mom, it's getting late ... what's for breakfast ... get up and get crackin' we're HUNGRY, you can't stay in bed all day!"
The most obvious example of a dog concerned about me when I was sick was back in '91 when I was spayed.
After I got home from the hospital I was in mega pain and also had an intestinal blockage and threw up whenever I tried to eat or drink anything (throwing up after major surgery is agony) ... it wasn't treated for over a week when they finally put me back in the hospital. Anyway, I had Echo, Yukon & Too stay boarded at the kennel and had the kennel bring Tex home since I could only care for one dog.
Tex was a bullheaded dog, I always called him a cement brain. He was always the center of his own world, only cared about himself. But this time was different.
When I dropped my shoe trying to put it on, Tex would pick it up and hand it to me time after time until I could finally get it on my foot ... he had NEVER been trained to do a take it/hold it/give, he just did this naturally.
When I needed help getting up, Tex, who had major juvinile arthritis, stood by my side so I could use his back as support to stand up!
When I dropped something on the floor he was Johnny-on-the-spot to pick it up and hand it to me.
Whatever I needed he was right there ready to help. After he finished eating his meal, he'd pick up the pan and hand it to me so I could wash it.
When I filled the water bowl I had to stand straight up and pour it into the bowl using a pitcher and then throw down a rag down and wipe up the spilled water with my foot ... Tex would then hand me the wet rag when I was thru.
Once I got well and back on my feet, Tex went back to being the same old Tex ... he'd look at me if I asked him to pick something up, give me the canine version of the bird, get up and walk off.
FWIW, despite the fact that he was extremely intelligent, it took us until he was over seven years old to earn his AKC CD title.
But when the chips were down, he was there for me and amazed me that he knew so many things to do that he would never do at any other time in his almost 13 years of life!!!
He was a cool dog!!!