There are other risks involved in a live vaccination, in particular the combined vaccines like MMR.
By the simultaneous administration of multiple vaccines, especially live vaccines there is the risk that an induced immune system (through the vaccine) causes a persistence of a virus in the body without eliciting an immune response to the virus. This virus can then possibly lead later to a chronic disease.
For example it is impossible that you have two viral disease at once(e.g.measles and mumps, I am not talking about Hepatitis B and C). You might have measles and afterwards you get mumps, but not at the same time.
Maybe this is of some help