When I was a child I had the most horrible migraines, I couldn't stand to have anything touching me, not clothes, sheet anything. I would often lay on the bathroom floor near the tolet were it was coolest and my dog, a mut that looked like a bearded collie, would come lay near me. She usually preferred to right on top of me but during these times she somehow knew and would always lay a close as she could without to touching me. She wasn't actually allowed in the bedrooms or the bathroom but when someone was sick she would do that belly crawl into the room. After my mother had surgery she would go into their bedroom and lay at the foot of the bed on the floor and only left to go outside and to eat.
Sir, my cat always knows when I'm down and will come snuggle with me.

Yes I believe animals have a way to sense these things, whether of not they act on it depends on the animal its self. Sometimes I think humans would be better at understanding each other if they talked less and used other methods to sense things instead.