My favorite story to tell (favorite in a horrible way) about these kind of issues is this: After an accident ~15 years ago, I had an open wound that wouldn't close because it was infected with MRSA which had lodged itself in the hardware that was screwed to my tibia. I was wearing a wound vac--a device that was attached to the open wound and maintained a constant suction to pull out any bad stuff. It was there to keep the infection from spreading long enough for the bone to heal so they could remove the hardware. I had a weekly doctors appointment at the local hospital check on the condition of the wound and change the dressing on the vac.
I was very careful. The doctor was in network, the hospital was in network, and my copay was, I believe, $30 for each visit, covered under my insurance. One day I showed up for my appointment, they led me to a room where I waited 10 minutes. Then the doctor came in, and it wasn't the guy I had been seeing for the last three months. He told me my doctor had taken ill and he was covering the appointments. He asked me how I was doing, and had the nurse redo the dressing on the wound vac. Then we were done.
A month later I received a bill from this doctor for $1900 because he was out-of-network, and while my insurance covered out of network at 80%, they covered it at 80% of the in-network rate. The doctor billed out of network patients at about twice the negotiated in-network rate (a fairly common occurrence), so the 80% coverage was actually 40% coverage, and I owed the other 60% which came to about $1900 for 5 minutes of work (the nurse who changed the dressing worked for the hospital so her work and materials were covered in network.) And the kicker was that I didn't know a different doctor was going to be attending me until he walked in the door.
If you think that maybe your "max out of pocket" will save you, it won't. If the insurance lists a "out of network max out of pocket" it only covers expenses at a negotiated in network rate. Any amount your out-of-network doctor bills that is over the insurance company's in network rate is effectively unlimited. Despite having a $2500 in network max, ad a $5000 out of network max, and a very good health plan. I left that accident with about $35k in medical debt.
(An interesting side note, when it came time to remove the hardware, I went back to the same hospital/trauma center and doctor who had installed it--though a different hospital from above. A day before the surgery I went to the hospital for the pre-op stuff. At the end of the process, the hospital admin met with me to take my payment. I was expecting a $30 copay or similar, but I found out that while the hospital was in network six months before, they had dropped their contract with my insurance company a month prior and the pre-op was going to cost me $800. So of course I cancelled the surgery, but I was still on the hook for the pre-op work. Today every time I visit a medical professional of any kind, no matter how many times I've visited them before, I ask about in network coverage.)