Legally, a subsequent premium differs clearly from the first premium. If the first premium is not paid, the insurer can still withdraw from the contract. For subsequent premiums, § 38 VVG sets out the procedure: a formal reminder with a payment deadline of at least two weeks. After that, you risk losing your claim to benefits for insured events during the period of arrears.
The full procedure of § 38 VVG, including the option to cancel under its paragraph 3, only applies to insurance that is not compulsory, for example supplementary insurance. For comprehensive health insurance, the special rules of § 193 VVG take priority. Contrary to a widespread error, § 39 VVG does not govern cancellation. It governs the premium settlement when a contract ends early.