[WalterBright]

Get the estimate in writing and then compare it with your bill. If there are more charges, they'll need to explain how those are unforeseen factors, or you can legitimately refuse to pay for them.

You can also show the estimate to other hospitals - this will often loosen them up about quoting a price because they'll want the business.

[FireBeyond]

Uh, what? There is nothing binding about the estimate (unless you're in the auto repair industry), so you cannot "legitimately refuse to pay for it" unless they explain "how those are unforeseen factors".

Typically, indeed, you'll find wording to that exact effect. "This is only an estimate. Actual costs may differ."

[WalterBright]

Ask for a binding estimate.