For example, the vaccine used for the 2009 Swine Flu:
https://en.wikipedia.org/wiki/Pandemrix
"Pandemrix was found to be associated with narcolepsy from observational studies, increasing the risk of narcolepsy by 5-14 times in children and 2-7 times in adults. The increased risk of narcolepsy due to vaccination in children and adolescents was around 1 incident per 18,400 doses."
If you don't have a large section of the population unvaccinated, you can't conclusively prove that the vaccine was the cause.