New car: 30.11.?

Hello, a case from the family: In August 2013 a new car was insured by telephone with a company (multiple test winner), which however proved to be very unreliable, e.g.: – The insurance confirmation (liability obligation) was not sent on time to the registration authority, so that the authority called the policyholder and exhorted. Only numerous telephone calls moved the insurance to send the confirmation to the authority in the short term. – The telephone call The insurance policy and the calculation of the contribution for this purpose were not sent until October, despite repeated inquiries. The question arises as to how the processing should take place when a damage occurs. Therefore, the policyholder wants to terminate the insurance on November 30. However, the insurance company refuses, on the grounds that the insurance company is not obliged to pay the costs. I know that nobody knows the contract here, but can that be true? Are there such clauses and are they legally permissible? Thank you very much, Marie-Luise