It is possible to run low on vitamins (depending on how long you go). If it's just 3-7 days, then it doesn't matter as much, but some people go a full 28 days (you can theoretically go about 40 days without causing problems, but as it will take another 40 days to break the fast, it proves to be too much for someone's first time (with 3-7 days working better)).
The step in a water fast that does most people in is the breaking of the fast. The key to a successful water fast is properly breaking the fast. It can be even more difficult, as while any hunger usually goes away within the first day or two, when the fast is first broken (with your first taste of 50% watermelon + 50% water, for example), it will kick the digestive system back into action, and you may easily become overwhelmed by an intense hunger. If you don't have the willpower to ignore this hunger, then you may end up eating something prematurely or eating too much.
The solution that works for most is to either buffer up on vitamins and minerals in the weeks leading up to the fast (which also ensures they don't have any hidden deficiencies), or to start with a juice fast (vegetables blended in a Vitamix, for example).