Yes, I've done this. I paid an out-of-network specialist for a second opinion on a cancer treatment. My HMO ended up following that specialist's recommendation. While I likely would have survived either treatment, this one was shorter and had fewer long-term side effects. I paid for the consultation out of pocket. It was less than $1,000.

Not that outcome matters much to your question, but I have fewer scars, more original body parts, and better overall body function than I would have had with my HMO's original plan.