It was discovered only really with the work of Siegel that Riemann made his hypothesis after computing the first few zeros and seeing they were on the critical line. In many ways the complexity of the zeta function is controlled by log log log T, and some heuristics say we shouldn't expect to see counterexample until height around e^e^e^3. Here's one example. The function S(T) measures the difference between the number of zeros up to height T and the number asymptotically expected. So in particular it jumps by 1 whenever there is a zero. If there's a zero off the critical line then there will be two zeros symmetric around the critical line, so S(T) would jump by 2. The biggest value of S(T) seen is around 1.6 so in a very real sense there isn't "room" for a counterexample just yet.