Wrong, wrong, wrong... you can not get measles from the measles vaccine. The MMR is a live, attenuated vaccine, so there is no concern about transmitting a disease-causing virus from a recently vaccinated individual.

Here's some actual information on MMR vaccine from Oxford University. It has the good and the bad.
http://vk.ovg.ox.ac.uk/mmr-vaccine

Herd immunity is a fairly simple mathematical formula, and simple concept to understand. One dose of MMR vaccine is 93% effective against measles, 78% effective against mumps, and 97% effective against rubella. And yes, there will be people that will still catch measles, even after having received the vaccine. In a vaccinated community, the majority of people will not catch measles and will limit the spread in the population. In an unvaccinated community, the majority of people will catch measles and will spread it to other communities.

Here's a simple diagram to help (in section B you can see the other "susceptible" individuals are protected.):

and the paper if you would like to read it - https://academic.oup.com/cid/article/52/7/911/299077

Oh, and you might be right. If you are an adult, have your measles titers tested to see if you are susceptible (even after having received the vaccine as a child). If they are low or no longer present then get a booster.