Good question! The reason is that no one can be sure that the pup will actually mount a good antibody response to the disease just from one vaccination. The age of the pup and just how much immunity it has received from its mother will complicate the "probability of protection". So a puppy that has "borrowed" a lot of imminuty for the mother during early nursing (called passive immunity) will limit a vaccine's effectiveness.
The idea is to get the vaccine into the pup as soon as the mother's passive immunity wears off so that the pup can make more lasting immunity of its own. (Consequently, this is also the time when the pup is most prone to diseases.) The precise time when a pup can respond well to a vaccine is variable -- it might occur at 6 weeks or perhaps 12 weeks. So why not begin vaccinations at 6 weeks and end them at 16 weeks?
On very rare occasions your puppy may have a reaction to a vaccination. If your puppy has any trouble breathing after a vaccination, or seems weak, staggers, has pale gums or seems at all unresponsive, contact your veterinarian immediately! If, however, your pup simply seems a little tired or slightly uncomfortable where it was vaccinated, that is an entirely different and mild response to the vaccination. When in doubt, call your vet.
Image: Jerry / via Flickr