I think the above posts have done enough to highlight how difficult becoming the best really is, and I agree that without insane natural talent getting that good is extremely difficult, but I genuinely believe most people can become, if not the absolute best player, at least one of the top 100 with enough perseverance, drive, and above all, a willingness to learn from one's mistakes. I think the last one is probably the most difficult, but if you have the mindset that every defeat is just an opportunity to learn and improve, and if you can observe the mistakes that will cost you games and focus on fixing them, I think you'll keep improving.
It takes a lot of humility, and the #1 thing I notice that keeps people from improving is when they don't try new things after a loss, when they keep making the same mistakes but are hoping for a better outcome. Nothing you do in Melee is sacred, and if it isn't working for you, whether against a particular opponent or just in general, don't be afraid to try something new. If the new stuff works, keep doing it, and if it doesn't, forget about it. It's all about listening to negative feedback and making adjustments as necessary.
tldr; Getting good is hard, learn from your mistakes and you can get pretty good, imo.