Occasionally, a driver will ask you to handle a car accident without going through insurance companies. This can seem like a practical option if the accident was minor, and the driver is willing to pay for damage to your vehicle. However, if you agree not to file a claim, you could be enabling a driver who really shouldn't be on the roads.
Drivers who don't want an accident reported to the insurance company don't usually have a good reason for doing this. Yes, perhaps that driver doesn't want their rates to go up over a minor accident. However, if you are dealing with a driver who has a criminal history, or a bad driving record, there is no reason you should do them favors.
If you have agreed to keep an accident "off the record," nothing is keeping you from filing a claim later on. You are not obligated to keep your verbal commitment to the at-fault driver, but you should consider the timeliness of your claim. Your insurance company will likely question why you waited to report the accident. They may question your claim more than they would have if you reported the accident right away.
Most insurance companies expect you to seek medical treatment shortly after a car accident if you have been injured. They rarely believe in "delayed injuries," so if you suspect you have been hurt in an accident, you are better off getting medical care as soon as you can. First, you should be concerned with your health. Second, you don't want the insurance company to assume your injury is not as serious because you waited.
Keep in mind; some insurance policies may require you to report any accident to the insurance company. By withholding information from your insurance company, you could be putting yourself at risk.
By agreeing with the other driver to not file a report, you are also putting yourself at their mercy. At the scene, they allege they will pay for damages to your vehicle, but when that bill comes in, they could easily avoid paying. If that is the case, you will have to report your accident to the insurance company just to get repairs to your car. This is a common outcome when the other driver asks you not to file a claim.
Finally, consider the ethical implications of keeping a secret for the other driver. Mistakes happen, and sometimes someone drives when they shouldn't. However, if they want to keep information from their insurance company for criminal reasons, you are allowing them to continue to take risks on the road.