With an inactivated (killed) vaccine, it isn’t possible. Dead viruses or bacteria can’t cause disease. With live (weakened) vaccines, some children get what appears to be a mild case of disease (for example, what looks like a measles or chickenpox rash, but with only a few spots). This isn’t harmful, and can actually show that the vaccine is working. A vaccine causing full-blown disease is extremely unlikely.

Source: CDC: Parents’ Guide to Childhood Immunizations FAQ