It depends entirely on what damaged it. A vehicle strike, a storm driven object, or a tree limb is sudden accidental damage and is commonly covered, subject to your deductible, which on a $250 to $800 panel job may exceed the repair. Wear, rust, a tired spring, and a worn out opener are maintenance and are not covered anywhere. Photograph vehicle or storm damage before anything is moved, and do the deductible math before opening a claim for a small job.
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