I would like _someone_ who believes that he could deal with any illness just using his mind to suffer from one of these illnesses
Maybe that's why those people don't suffer from those illnesses in the first place.

Sorry to disappoint you, but a pretty famous austrian former physician did. I can't remember the name, but he convinced the parents of a 7 year old girl to deny medical treatment and to follow his rules to help her subconcious to fight the cancer down. When a judge finally ordered that the child was to receive treatment, it was too late. Well, sarcastic as life is, that austrian man died of cancer himself, about 3 years later.
But generally: You simply don't hear much about these people, and this is especially true when they actually become ill.

I'm not saying deny medical treatment, that wouldn't make any sense. A man can go over a month without food if he's determined, but if the food is there then eat it.

Well, what I am usually saying is that if god didn't want us to use machines or medicine, he wouldn't have given them to us (by making us able to invent them).

Well the reason I asked is because Jesus said you could move mountains if you have enough faith.
If you ever care to investigate the circumsances under which he said that, you will find out that he was talking in his usual pictures and was also talking about changing other peoples minds. I can't find the exact place in the bible where he said that, so this is all from (limited) memory.

cu,
sven