TITLE: Theorems that impeded progress
QUESTION [82 upvotes]: It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples:

Faber's Theorem on polynomial interpolation: Interpreted as saying that polynomial interpolants are useless, but they are quite useful if the function is Lipschitz-continuous.
Squire's Theorem on hydrodynamic instability: Applies in the limit $t \to \infty$ but (nondimensional) $t$ is rarely more than $100$.


Trefethen, Lloyd N. "Inverse Yogiisms." Notices of the American Mathematical Society 63, no. 11 (2016).
  Also: The Best Writing on Mathematics 2017 6 (2017): 28.
  Google books link.

In my own experience, I have witnessed the several negative-result theorems 
proved in  

Marvin Minsky and Seymour A. Papert.
  Perceptrons: An Introduction to Computational Geometry, 1969.
  MIT Press.

impede progress in neural-net research for more than a decade.1

Q. What are other examples of theorems whose (correct) proofs (possibly temporarily)
  suppressed research advancement in mathematical subfields?


1
Olazaran, Mikel. "A sociological study of the official history of the perceptrons controversy." Social Studies of Science 26, no. 3 (1996): 611-659.
Abstract: "[...]I devote particular attention to the proofs and arguments of Minsky and Papert, which were interpreted as showing that further progress in neural nets was not possible, and that this approach to AI had to be abandoned.[...]"
RG link.

REPLY [5 votes]: The Mermin-Wagner theorem (1966) proved that for two-dimensional models with a continuous symmetry there was no finite temperature transition to a phase with long-range order via spontaneous breaking of this symmetry.
My understanding is that this was generally interpreted as proving that "two-dimensional models with continuous symmetries do not have phase transitions". I do not know for sure that this impeded progress, but I suspect that as it appeared to be apparently a no-go theorem this would have prevented researchers from looking for examples of phase transitions.
Remarkably, not too long after the theorem was published, Kosterlitz and Thouless (1973) showed that the two-dimensional XY model does indeed have a phase transition with diverging correlation length, but no long-range order. The mechanism was completely different (due to topological effects), and so managed to avoid the apparently hard constraints imposed by the theorem.