Yes. Many banks will give you the option to update your current card. There is one caveat, though: They will probably only do this for a fee.
Additionally, your bank may not offer very many design options, if it can offer them at all. For example, your current card may not be available in any other design. In cases such as these, your issuer may suggest going with another product, which may not include all of the benefits your current card grants you.