They (hospitals, out of network specialists) are also under no obligation to negotiate with you. You are on the hook for whatever your insurance refuses to pay, which can be substantial regardless of what your deductible is.

They might negotiate with you, but they might not, in which case you are at the mercy of whatever your state's creditor laws are. Florida? Great laws protecting your primary residence and wages from garnishment if you're head of household. Missouri? Not so much.