This could be the case, e.g. if you want wo deploy your model for object identification like a cat. Then this should manifest in your test set, too. Then the test set could be tailored to the business problem we want to solve - in our example: cats! It’s important to use new data (e.g. pictures of cats) that the model never saw before. Still the training data could also include other trainings examples (of cats but also other related animals).
My take: not necessarily! Adding new training data could help the model to learn abstract patterns better (like paws or so [which is definitely relevant to our cat example but many other animals as well]). So there are many relevant data and characteristics to be learned from other pictures w/ animals like tigers, lions, leopards, … 
, that could help to learn relevant features to identify a cat more accurately. So, the model could learn how edges and contours make a „paw“ or „whiskers“ or other features that are important to identify a cat and low level features like edges are hierarchically combined and enhanced to describe more advanced patterns to finally form objects, which contribute to the classification if we see a cat on the picture or not. This thread might be worth a look: What makes the different neurons in a layer calculate different parameters? - #7 by Christian_Simonis
Hope that helps!
Best regards
Christian