Short answer is, yes it's possible.
That is the very purpose of having a template hierarchy. Different templates having different functionality and design based on your needs.
For example, you can create a
tag-facebook.php and make it look like Facebook, while having a
tag-twitter.php that looks nothing like Facebook, but just like Twitter!
There is one thing you should consider. Most of the page templates have the following structure:
// The page's content is here
If you follow this structure, you can't fully customize your page since
get_footer() will always include the same header and footer templates in your page.
Unless, you define different headers and footers to use in different designs.