Burn-in has been proven effective in identifying and removing defective products before they are delivered to customers. Most existing burn-in models adopt a one-shot scheme, which may not be sufficient enough for identification. Borrowing the idea from sequential inspections for remaining useful life prediction and accelerated lifetime test, this study proposes a sequential degradation-based burn-in model with multiple periodic inspections. At each inspection epoch, the posterior probability that a product belongs to a normal one is updated with the inspected degradation level. Based on the degradation level and the updated posterior probability, a product can be disposed, put into field use, or kept in the test till the next inspection epoch. We cast the problem into a partially observed Markov decision process to minimize the expected total burn-in cost of a product, and derive some interesting structures of the optimal policy. Then, algorithms are provided to find the joint optimal inspection period and number of inspections in steps. A numerical study is also provided to illustrate the effectiveness of our proposed model.