Non-mutal exclusive classification task examples
For example, due to the complexity of the images in the ImageNet database. Algorithms will often use hundreds or thousands of output nodes to be capable of classifying a large array of different things. Researchers also relax the cost function and allow the $k$ highest outputs to be considered. If one of these is correct then the example is considered to have been correctly classified. Furthermore with the existence of such complex data, often times a certain object might be a subset of another object. 

For example, consider the picture of a human being. If the algorithm detects a human with the highest activation at the output node, but also has high activations for hands, eyes, face. This is not wrong! The image does in fact contain these objects as well and them being detected by the model is encouraging.