Some dogs can predict when a child will have an epileptic seizure, and even protect the child from injury. They're not trained to do this, they simply learn to respond after observing at least one attack.
Dogs can also indicate to their diabetic owners when their sugar rate is too high or too low, just by smelling them; after they saw this particular state leading their owner to a crisis one or two times, they learn to adopt a "warning behavior" that the owner will recognise, and they do not need any special training for that. They'll even awake them up if somethig like this happens in the middle of the night.
They can also do this with people that are narcoleptic; these people somethimes fall asleep while standing up and thus can hurt themselves pretty badly. (A lot of narcoleptic people come to have a boxer-looking face because of that. <img src="/ubbthreads/images/graemlins/exclamation.gif" alt="" /> imagine if they fall asleep while they're going up or down some stairs <img src="/ubbthreads/images/graemlins/eek.gif" alt="" /> <img src="/ubbthreads/images/graemlins/ouch.gif" alt="" /> )
A related fact: Narcolepsy is a genetic disease widespread among the doberman breed dogs; as soon as these dogs are getting a little excited, they fall asleep. It's very odd to see.