Were services required to be pre-authorized? If so, the clinic where prior authorization was requested should have received an authorization number. If the facility billed without that number on the claim, ask them to resubmit. Request that number for your appeal if you need it. Most often, clinics will request prior authorization on behalf of patients, but ultimately, the insured is responsible to make sure prior authorization is obtained and could be held responsible for the bill if that doesn't happen. Sometimes, due to a clerical error by the facility or by the insurance company, that authorization number may be missed on the claim. Sometimes it can be as simple as calling the insurance company with the number.
In the future, if there are tests/services ordered, I would check beforehand to find out for sure if authorization is required, make sure the office obtained authorization, and request a copy of the authorization number and the date authorization was received (you should also receive a letter from the insurance company with that information). You'll want that information if you ever have to file an appeal. Except in an emergency situation, I personally wouldn't have a procedure/test that requires authorization done until I knew for sure that it was authorized, not even if someone says they are "in the process of obtaining authorization" or authorization is pending.
On a side note, it can be a good idea to make sure of whether the physician/hospital, etc. is in network. I've seen denials where the hospital is in network, but the ED doc who saw the patient isn't, so the facility bill is paid as in-network but the doctor's bill is denied or paid as out of network with a higher deductible and patient responsibility. I find that infuriating - who has time during an emergency to check into such a thing? I've seen it with Physical therapy, too. The hospital was in network for inpatient visits, but outpatient physical therapy was not, resulting in an out-of-network fee. The insurance company and facility will both just say it's the patient's responsibility to know their coverage, as convoluted and confusing as that can be.
Have you received an EOB? That EOB should state the reason for denial if a charge is denied, and it will also include the patient's responsibillity. If it says the patient responsibility is $0, or only a portion of the $3000, I would go back to the hospital's billing department and refer them to the EOB. They have contracts in place with the insurance company and shouldn't be billing you for more than what the EOB says is your responsibility.
If the claim was denied, both you and the hospital should receive a denial letter from the insurance company explaining why it was denied and the appeal process. Often, it will also state whether a patient can be billed for the denied services. If you haven't received a denial letter but you were told it was denied, call and request a copy from the insurance company. If it's a medical necessity denial or a denial for lack of pre-authorization, work with the billing department and physician to get the documentation you need to support the test. Be mindful of timeframes listed in the denial letter. Ask if there is someone in the billing department or in the hospital who handles denials - that person may be able to assist you with the process.
Good luck, OP. I used to write appeal letters on behalf of a hospital and sometimes specific patients, and I know how confusing it can be. Hope it all turns out okay.