No, you cannot assign both the design and product templates, together for one product. Mainly because both the templates are used for different types of products and serve different purposes. Product Templates are used for products with fixed attributes. (calendar, visiting card, photo album etc.) The attributes of the product template cannot be changed by your customers. Whereas design templates are design ideas that help your customer to design their products. The attributes of the design templates can be changed, removed and added by your customers.
While designing a product, your customers will only be able to view one template option, in the designing section on the designing page.
Visit this article to find out the difference between a product template and a design template.