Recently we talked about the Riemann hypothesis in class, and yesterday I stumbled across the Goldbach conjectures. I realized there are quite a few theories that assert a particular property to all numbers of a specific form (greater than some n), which have been confirmed for the first gazillion numbers. Yet that is not sufficient proof. (Of course, some of them have been proven.)
So out of interest: Are there any such theories that remain valid for a wide range of numbers and then suddenly break off?