Which theorems were a disappointment when they were proved?

The obvious first answer is the incompleteness results of the early twentieth century, and the disappointment there was as large as mathematics has ever produced.

The programme at the time was to put all of mathematics on a complete, consistent, mechanically checkable foundation. What was proved instead is that any system strong enough to describe ordinary arithmetic contains true statements it cannot prove, and cannot establish its own consistency.

That closed the programme permanently. People had not merely hoped for the other answer; a great deal of work had been organised around the assumption that the other answer was forthcoming.

In hindsight the disappointment was real and the consequences were enormously productive. The techniques developed to prove it founded computability theory, which is a large part of why computers have a theory at all. The letdown and the payoff are the same result.

A more concrete one that let down navigators rather than logicians: you cannot make a flat map of the sphere that preserves both angles and areas.

This follows from the fact that a sphere and a plane have different intrinsic curvature, and curvature is preserved by any distance-preserving map. So every world map is a compromise, forever, and the argument about which projection is best is permanently unresolvable because there is no best.

The disappointment here was practical and long-lived. Centuries of cartographers were essentially looking for something that had already been ruled out, and the theorem told them what kind of trade they were choosing between rather than how to avoid the trade.

One where the disappointment was about how rather than whether: the four colour theorem.

It is true — four colours suffice for any map. But the proof required checking an enormous number of cases by computer, and for a long time no human-readable proof existed. Many mathematicians found that unsatisfying in a way that had nothing to do with doubting the result. They wanted to know why it was true, and a verified exhaustive case check does not tell you.

That reaction has softened as computer-assisted proof has become normal, but it opened a question that is still live: what is a proof for? If it is to establish truth, the case check is fine. If it is to produce understanding, it is not.

From my end of the subject: the unsolvability results.

There is no general algorithm that decides whether two elements of an arbitrary group are equal. There is no general algorithm that decides whether a program halts. There is no general algorithm that solves arbitrary polynomial equations in integers.

Each of these was posed as a challenge — go and find the procedure — and each was answered with a proof that no procedure exists. That is a much colder answer than "nobody has found it yet", because it forecloses the search entirely.

The consolation is that all three led directly to understanding why certain problems are hard, which turned out to be more useful than any of the procedures would have been. Impossibility results are how a field learns where its boundaries are.