How long do you have to report an automobile accident to the insurance company?

Depends on whether it is YOUR carrier or the carrier of another at-fault driver. Auto policies state, in some form, under Collision and Comprehensive (sometimes now termed "Other than Collision"), that you must report the accident to them in a reasonable amount of time: the language can vary and can state "as soon as possible", "within a reasonable time period", "within 30 days", etc. They can and sometimes DO deny coverage if you've not complied with this. The reason they will deny is because a damaged car is more likely to sustain additional damage: rust, leakage, loose parts falling off, sometimes more prone to subsequent accidents (dependent on the type and location of damage). If you are NOT At Fault, but fail to report your damages promptly, or make your car available for inspection promptly etc, the insurance CAN deny your claim. They are not likely to if there is truly damage they are responsible for, but they can, and are within their legal rights to, reduce the amount of your claim if failure to repair your damage in a reasonable amount of time (called "mitigating" your damages) increases the amount needed to restore you to pre-loss condition.