Interesting question! And it's a sticky one, too, as the idea of “failure" isn't always clear-cut. If your character sets out to win his beloved's hand, but ends up realizing something else is more important, has he succeeded or failed? Additionally, a character who always succeeds may, in fact, find his conflict in that success when his wife becomes envious and angry.
While a character may or may not fail, he should be human and, as a result, flawed. Like humans, characters succumb to their weaknesses, get in their own way, and misunderstand. They want something they cannot have, flash anger when they should embrace, carry guilt when they should forgive themselves. This quality creates authenticity and conflict.
Consider Jay Gatsby in F. Scott Fitzgerald's novel The Great Gatsby. He throws lavish parties. People admire him and find his mystery exciting. His library is well stocked and his closet is filled with expensive clothing. But Gatsby isn't all that he seems. He throws these parties only to cross paths with his long-lost love, Daisy. Many of the books in his library are unread, the pages still uncut. When his car strikes and kills a woman while Daisy is at the wheel, he takes the blame for it. These complexities heighten the conflict. Who is Gatsby? Will the façades he's created remain intact? What will happen if they don't?
Or consider Shukumar in Jhumpa Lahiri's short story “A Temporary Matter." His relationship with his wife, Shoba has been strained since the loss of their baby. Shukumar is saddened by the distance between them, yet he contributes to it. He sets up his office in the room that was to be the baby's nursery. He stays there to work during dinner, avoiding Shoba. When the power company turns off the lights and they are forced dine together several evenings in a row, they begin to reveal secrets they've kept from one another over the years. Shukumar is both excited by this time together and put off by it, frustrated that he cannot retreat to his home office. The inner push and pull that Shukumar experiences gives these truth-telling sessions tension.
A character's failure can certainly be an important part of a fiction's conflict. But compelling conflict is often found in the nuances of characterization, which aren't always so straightforward and easy to label.