I've heard from anti-vaccine advocates that a number of diseases were on the decline, and that vaccine use was insignifcant in that decline. But the vaccines were introduced anyway so you could say it was the vaccines that did it but other factors were at hand.

Sometimes I think correlation not only doesn't equal causation, but counters causation, like a son doing well not because of his overbearing mother, but in spite of it. (She would claim the credit anyhow.) - I've read about vaccines causing diseases they are proclaimed to guard against - "weaponized vaccines."