Is it very common to have both hypothyroidism and fibromyalgia?

Possibly. However, treatment of hormone abnormalities would have to be done before calling the symptoms fibro. Normalizing the thyroid could definatwly help with the pain.
Yes. Yes, there is an increased incidence of hypothyroidism in people with fibromyalgia. In addition fatigue and pain are symptoms of both fibromyalgia and hypothyroid. Anyone with symptoms of fatigue should have their thyroid tested.