Named entity recognition (NER) systems have seen rapid progress in recent years due to the development of deep neural networks. These systems are widely used in various natural language processing applications, such as information extraction, question answering, and sentiment analysis. However, the complexity and intractability of deep neural networks can make NER systems unreliable in certain circumstances, resulting in incorrect predictions. For example, NER systems may misidentify female names as chemicals or fail to recognize the names of minority groups, leading to user dissatisfaction. To tackle this problem, we introduce TIN, a novel, widely applicable approach for automatically testing and repairing various NER systems. The key idea for automated testing is that the NER predictions of the same named entities under similar contexts should be identical. The core idea for automated repairing is that similar named entities should have the same NER prediction under the same context. We use TIN to test two SOTA NER models and two commercial NER APIs, i.e., Azure NER and AWS NER. We manually verify 782 of the suspicious issues reported by TIN and find that 702 are erroneous issues, leading to high precision (85.0%-93.4%) across four categories of NER errors: omission, over-labeling, incorrect category, and range error. For automated repairing, TIN achieves a high error reduction rate (26.8%-50.6%) over the four systems under test, which successfully repairs 1,056 out of the 1,877 reported NER errors.