Yes you can. You can claim your no fault or accident benefits and you may also be able to sue the driver of the other car.
Just because the insurance company says both drivers are a fault that doesn’t mean that the insurance company is correct.
The insurance company determines fault according to the fault chart. The fault chart is only meant to get it generally right most of the time. For example the chart does take into account circumstances in which the accident occurs such as road conditions, visibility or other cars involved in the accident.
In a law suit it is the judge and jury who will determine fault. The opinion of the insurance company is not even evidence in the law suit. So even though your insurance company finds you partially at fault, at trial the other driver may be held 100% responsible for the car accident.
Even if the insurance company is right and you are partially at fault for the accident you can still sue. The amount of your damages will be reduced by the percentage you are at fault. For example if you are 25% at fault you will only get 75% of your claim.