The Question Asked
• When, in the history of mathematics, have problems "like P vs. NP" arisen and then been solved?
• In those cases, what were the resolutions?
The Runtime Fence Problem for TMs Given a Turing Machine (TM) promised to be in P, and a non-negative real runtime exponent $k$, a commonplace and eminently practical math-and-engineering question is this: "Is the TM's runtime $O(n^k)$ with respect to input length $n$?"
We call this is the Runtime Fence Problem for TMs.
The Relevance to the Question Asked We imagine that "an invisible fence" separates TMs whose runtimes are slower-than-$n^3$ from TMs whose runtimes are $n^3$-or-faster, and we are asked to decide whether a given TM resides on one side or the other.
The Resolution Emanuele Viola has proved that the Runtime Fence Problem for TMs is undecidable.
So in regard to runtime exponents, the "invisible fence" turns out to be truly invisible.
Natural Extensions
The Runtime Fence Problem for Languages Given a language L, the Runtime Fence Problem can be posed for the most efficient TM that recognizes that language. We call this The Runtime Fence Problem for Languages.
The Resolution The Runtime Fence Problem for Languages is natural, open, apparently difficult, and conjecturally undecidable.
For definitional details, comments, and mathematical history, see the TCS StackExchange community wiki "Does P contain languages whose existence is independent of PA or ZFC?."