You cannot develop Asperger's or any type of Autism late in life. One of the absolute diagnostic criteria to be able to get a diagnosis of any form of Autism/Asperger's is that you have to prove that the traits and symptoms were there from early childhood, like toddler age or younger. Now there are many things that can cause sensory sensitivities. You can have or develop sensory sensitivities and not have an Autism Spectrum Disorder. So you can develop sensory issues later in life, but it won't be because of an ASD. But you can also have an ASD and not have sensory issues even though that is more rare. So it is possible that you have ASD already and that your sensory issues developed as you got older. But ASD is something that you have to have had since birth or at least proved to have had in early childhood so no, you cannot develop it or catch it as an adult or older child. There are other conditions that you can develop as you are older that may have similar traits as ASD but they will not be ASD.
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"I'm bad and that's good. I'll never be good and that's not bad. There's no one I'd rather be than me."
Wreck It Ralph