I do, too.
But even without feeling they can be "psychic" or whatever, it makes total scientific sense if you know how a cat thinks/lives. Cats have a much more sensitive sense of smell than we do. When you're stressed, your body produces a different smell as it releases some other kind of chemical. I don't remember all the technical terms, but the cat can literally smell your fear, uncertainty or tension (or on the other end, happiness, love, etc). Furthermore, we're talking about animals who primarily communicate via body language. Watch them sometime, and you'll see that in their stance, ear position, whisker position, eyes, etc. Being so attuned to body language, they can pick up on your tension as well even without the further chemical release. And in the animal world, when one member of the clan/tribe/group is tense, it usually means trouble and the other members will respond either in kind, by hiding, or by turning aggressive. As far as they're concerned, we aren't humans, we're just strange looking cats and part of their "clan". So it makes total sense that the calmer you stay and more confident you are, the more likely your animal is to respond in kind.
If you do believe your animals can pick up on your thoughts or intentions, then it's also important to think good thoughts in your dealings with him. Tell him how brave he is, and encourage him that he'll get better. The more you believe it, the more he will, the better his chances will be that he'll improve. Also tell him how special he is to you and how much you love him. Shoot, even if you *don't* feel they can pick up on those thoughts, what harm does it do?
I feel that whether they pick up on thoughts, some chemical clues, body language, or some other method we don't yet understand, cats (and dogs, and for that matter to some degree *FISH*, or so I've seen in my Bettas) *do* pick up on our emotions and physical/mental states. They know when you're happy, sad, loving, or hateful. And just like humans, positive thought can help increase the chances of recovery.