Depends on the person - but it is conceivable for a human being to go without sleep for two weeks. Although you will probably start to hallucinate at that point.
When I was in the army we were told that we needed at least 4 hours continuous sleep in order to function properly. We never got more than 6 hours because we had to take shifts on watch. I still managed to do a full day's activities on 4 hours sleep a day, albeit with slower reactions times - but i could still run, shoot and shout out orders.
If you are suffering from insomnia or sleep deprivation the first symptom is usually slurred speech, reduced reaction times and a lack of creative thinking. Severe sleep deprivation (greater than 24 hours) can result in hallucinations, mood swings and a weakened immune system (although it should recover in time) . Prolonged lack of sleep - less than 4 hours per day may result in weight gain, reduced mental agility, sickness and possibly death.
Interestingly our memory actually improves when we are deprived of sleep! Although we are unable to learn anything new without getting at least 8 hours sleep - 8 hours being the minimum amount of time before the brain goes through a complete sleep cycle.
It should also be noted that after a prolonged period without sleep the brain will begin to shut itself down and go into a state called "microsleep" - your eyes are open and you appear awake but the brain goes in to sleep activity. But this is different from actual sleep since you do not enter full REM sleep, its only the early stages of sleep which shuts down certain parts of your brain.
Various white papers, my own knowledge.