Happens 1 in 100k people every year. It's called Guillain-Barré syndrome [1]. After an infection your immune system gets confused and starts attacking your peripheral nerves.
Yes, of course there are tons of autoimmune diseases (from mild allergies, to lupus, multiple sclerosis, and sympathetic ophthalmia), but Guillain-Barré syndrome is an autoimmune disease that's known to occur after otherwise banal infections or even vaccine shots. That's why I considered it more relevant to the OP than other autoimmune diseases.
[1] https://en.wikipedia.org/wiki/Guillain–Barré_syndrome