Generally, yes. If you were hit by a car on your bike and the driver was at fault, you can file a claim with that driver’s insurance company. You have to prove the driver was at fault. Then you can negotiate a reasonable and fair payment for your injuries and losses from that insurance company.
Insurance negotiations are complex affairs that are often best left to attorneys. But filing a personal injury lawsuit is usually a last resort where the insurance company is low balling your claim or is denying it outright.
However, bear in mind there are cases where the bicyclist could be partially at fault for the accident. If that is the case, you may not be able to recover the full amount of damages you would have otherwise. Most states reduce the amount of damages you can receive based upon your percentage of fault for the accident. For example, if you are hit by a car and suffer $2000 in injuries in medical bills and lost wages, but were 50% responsible for the accident, you may only receive $1000 in compensation.