Can an auto insurance company deny a claim?

I had an accident last week for which I was not at fault. The police department clearly agreed with me and they gave the traffic ticket to the other driver for reckless driving. I filed a claim with the other driver's insurance company and they say they are investigating the accident. I asked how long it would take for them to fix my car and they said that if they thought they were responsible they would pay for it, otherwise they would not cover the costs of repairing my car; which are pretty elevated. Can they do this? I have a police report which clearly states that the accident was not my fault. What should I do if they deny my claim?

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