I think that it's definitely possible to go far with competing if you work hard and put your mind to it. For Olympic level/international level though...realistically, how many people have the horsepower to back that? Horses with the talent -and- mindset for that level of work and competition are very few and far between. They aren't cheap when you find them either (usually). Certainly a middle, run of the mill and good hearted horse will take you far (my pony took me 4th level in dressage and was bought for a whopping $450) but for elite high level competition riders will require horses that aren't every day horses; those horses are beyond spectacular. Also, riders have to have some sort of talent to back it. Some people work and work at their riding, are dedicated as all heck, and will still never make it to those international levels, but, some will. I honestly feel riders need to have some natural talent and feel to back their passions in order to make international level. And lessons, lessons, lessons...you absolutely cannot make it their without a coach. So, money isn't everything but it can influence a lot. You need the money for the horsepower and the trainers. I'm sure this isn't the rule but it's probably more the norm. Granted sponsers play a huge role for many high calibre riders (but that's when you have to be talented enough to catch someone's eye)
That being said: Follow your passions and never give up. Hard work will pay off whether you want to ride at a local, national, or international level. You don't know unless you try.