Your brain will never say " ok i give up." Its counterproductive to the whole procreation of the species. It is theoreticly possible for one to go into shock and die. if someone went into psychosomatic shock ( i dont even know if it exists though, bust sounds like it could) they could seem to just die. But your brain would have to persieve a HUGE threat to get you to that stage. Plus this would be all/mostly subconscious so you probably wouldn't be aware of it.
More likely you would die due to complications brought on by prolonged stress. General Adaption Syndrome(GAS) states that the body exists at homeostasis ( a baseline norm) untill a threat is percieved. then the body goes into an alert phase when your body pumps out tons of different hormones and chemicals ( epi, norepi, corticoids, etc.). Now this goes on for a few min ( 10ish iirc)
then the body goes into resistance phase. the chem/horm. are still being put out, just in lower levels. The longer this state lasts the more damage is done ( especially to the circulatory system, chem constrict blood vessels at this time and a lot more fluids are added ( aka smaller tubes need to carry more liquid then normal). this is were you encounter weakened systems ( immune, circulatory, etc).
then the body goes into exhaustion phase. now the chem/horm being up out drop fast ( to below norm lvls) and the body basically breaks down. organs fail. If you don't get out of this you WILL eventually die.
on another note, I'm enjoying my Stress and Coping class where we just learned the dynamics of a stress response.