Most likely, your car insurance company specifies in the details of your policy that you must give "prompt notice" of any car accident or loss. If you fail to promptly report a car accident and the delay prevents the car insurance company from investigating thoroughly or taking other action, the company may deny the claim outright or lessen the amount it pays.
There are a few important reasons you need to notify your car insurance company promptly in the event of a car accident. If you try to settle a claim yourself, you may do or say something that could make it difficult for the car insurance company to settle the claim later. This may jeopardize the terms of your policy, and the company may deny the entire claim or lessen the amount of money you receive. If you weren't at fault, reporting the accident to your insurance company will make them aware of the situation and they will take your side immediately, assisting you in negotiating anything regarding the settlement. Attempting to handle the situation yourself can be far more complex than you might expect, even in a straighforward case, as you'll likely be dealing with the other driver's insurance company, and it's in their interest to pay you as little as possible. Putting yourself up against that kind of negotiation is not pleasant or wise.
If you were at fault in the car accident, or even if you just suspect you may have been at fault, reporting the car accident is still highly recommended, even though people often avoid doing so because they fear the car accident will result in higher insurance premiums. However, if you don't report a car accident, the car insurance company will likely find out about the accident anyway, because the other driver (or his insurance company) will probably call them. It can also be considered an act of bad faith on your part to withhold this information.
It's also not always true that your insurance rates will increase after an accident. The best course of action is to submit the report immediately and keep the car insurance company updated if anything changes. If the car insurance company becomes involved in any negotiation regarding the case, they will be in touch with you as well, and the entire process will go more smoothly if you offer full disclosure along the way.
You may also wish to contact an auto accidents attorney, since your car insurance company is focused primarily on looking out for their own interests more so than making sure you are treated fairly.