If a vaccine is given when a baby still has antibodies to the disease, the antibodies can stop the vaccine working. This is why routine childhood immunisations do not start until a baby is two months old, before the antibodies a baby gets from its mother have stopped working.
This is also why it is important for parents to stick to the immunisation schedule, as a delay can leave a baby unprotected. A delay can increase the chance of adverse reactions to some vaccines, such as pertussis (whooping cough).
For more information, please check the NHS: Vaccination Schedule.