An example from set theory: My understanding is that it was once widely believed that all reals appearing in canonical inner models of large cardinals (at least up to supercompact cardinals) would be $\Delta^1_3$ in a countable ordinal. This is because it was assumed that linear iterations, the only kind known at the time, would suffice to compare such inner models. This assumption turned out to fail at the level of Woodin cardinals, far below supercompact cardinals. The resulting non-linear iterations (iteration trees) are a basic part of inner model theory today, whereas canonical inner models for supercompact cardinals are still far out of reach.
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