You win whether you receive a fellowship or not.
By completing the arduous fellowship application process, you will emerge as a winner in one way or another. Those who successfully complete final product gain valuable new skills to navigate the wider world and negotiate life.
The fellowship application process is a rite of passage into a more responsible and mature world where your efforts are truly respected by experts in your field. All applicants are honored. Every completed application is a proud achievement. Most importantly, the process forces you to engage in self-reflection and honest self-assessment, which teaches you a great deal about who you are and what you want out of life.
Further, applying for fellowships prepares you for what the graduate school application process is like. If you think graduate school is a possibility, apply for a fellowship first (or even at the same time). Learning to articulate an argument and perfecting the art of selling yourself are invaluable skills that will serve you as you continue on your career path.
You should disregard fellowship opportunities only if ineligibility has been established. Assume eligibility until proven otherwise, but by the same token, don't overstretch qualifications because fellowships are highly competitive. There is a fellowship that is the right fit for you, and sometimes it takes time to figure out which one is best for you.