BiLSTM has been prevalently used as a core module for NER in a sequence-labeling setup. State-of-the-art approaches use BiLSTM with additional resources such as gazetteers, language-modeling, or multi-task supervision to further improve NER. This paper instead takes a step back and focuses on analyzing problems of BiLSTM itself and how exactly self-attention can bring improvements. We formally show the limitation of (CRF-)BiLSTM in modeling cross-context patterns for each word – the XOR limitation. Then, we show that two types of simple cross-structures – self-attention and Cross-BiLSTM – can effectively remedy the problem. We test the practical impacts of the deficiency on real-world NER datasets, OntoNotes 5.0 and WNUT 2017, with clear and consistent improvements over the baseline, up to 8.7% on some of the multi-token entity mentions. We give in-depth analyses of the improvements across several aspects of NER, especially the identification of multi-token mentions. This study should lay a sound foundation for future improvements on sequence-labeling NER1.