yyyyyyy's answer is the correct short version.
There is only a single cryptographic algorithm that is mathematically proven secure: the one-time pad. It's hardly ever used because it's impractical: the key size is as large as the data to protect. You can prove that any algorithm that is secure against an adversary with infinite computational power is essentially equivalent to a one-time pad, and thus impractical in most situations.
Every practical cryptographic algorithm requires an assumption that some mathematical function cannot be computed in practice (because, for sizes used in practice, the best-known method require billions of computers to work for billions of years, or maybe even require more energy than exists in the universe). We believe that these functions cannot be computed faster because lots of very smart people have spent years of their lives trying in vain. It is, mathematically speaking, possible that someone will find a vastly faster method one day. We consider it unlikely because so many people have failed, but it cannot be ruled out.
A gradual break is more likely than a sudden break because scientific progress tends to make small increments at a time. Scientists are dwarves standing on the shoulder of giants (as Newton famously wrote, as did others before and after him). The popular image of a scientist suddenly becoming enlightened and coming up instantly with a novel, practical idea hardly ever happens. It takes time, and often multiple people, for ideas to mature. Before an actual cryptographic primitive is broken, it's extremely unlikely that there won't be warning signs, such as theoretical advances in a related topic, slight improvements in our understanding of the mathematical structure of the primitive that lead to small but still impractical reduction of computation requirements to break it, etc.
All of this applies to algorithms that have been vetted by the world's top cryptographers. An algorithm proposed by a person or a small team working on their own may well be thoroughly broken because someone else thought of something they didn't. When you involve a significant number of experts, the risk of “nobody thought of this” dwindles.