Is it always wrong to write a sentence fragment?

What a wonderful question! No, it is not always wrong to write a fragment. However, you have to consider who you are writing to and why you are writing. If you are writing a cover letter for a job and your audience is your future employer, please don't write a sentence fragment. It will look sloppy. Same goes for research papers, formal essays, business reports, official emails and any writing in which you must perform professionally. On the other hand, professional creative writing uses fragments for artistic effect. A fragment in a well-written novel, story, poem, memoir, or creative non-fiction essay can be powerful. Yet, to break a rule with style requires knowing the rules. I think it was the artist Picasso who first learned to paint proper portraits and then bent "the rules" to create his colorful, twisted, memorable portraits for which he is famous. He said that he spent his youth painting like an adult and his adulthood painting like a child -- but his work is masterful. So, learn the rules, write complete sentences, definitely write complete sentences in formal written work, but play with words too, exercise your creativity, and try writing excellent sentence fragments.