Jess,
It depends of which vaccination you are talking about. If it's flu vaccination, then it's no guarentee, since there are so many different species, and so the jab you are taking may not work for that species (and the healthcare specialists always predict the wrong species every year, so ended flu jab for nothing).
But if it's for vaccination like Hep B, Hep A, Chicken pox, Pneumococcal, rotavirus, etc, these viruses have either haven't been evolved so much or have been known very well. So the vacciation usually very effective, almost guarantee you won't catch the disease once you got the jab.
For your neighbour case, you have to find out which vacciation did the kid take and what causes the fever. Then you make the judgement. It's all scientifically make sense.