For over $130$ years people have been steadily looking for a resolution to the following problem: what is the maximum number of limit cycles for the system of differential equations $x'=f(x,y), y'=g(x,y)$ where $f$ and $g$ are any real quadratic polynomials, in $x$ and $y$. In the 1950's, two Russian mathematicians (Petrovskii and Landis) wrote a paper claiming that the maximum is $3$. People tried to understand the proof but found holes in it, and attempted to fix it up. The famous Arnold was skeptical about the possibility. In the 1970's, two Chinese mathematicians (Chen and Wang) discovered a specific set of coefficients for the polynomials $f$ and $g$ and showed that this system has $4$ limit cycles. Of course, nobody tried to patch up the previous claim of max=$3$.
I thought this to be an amusing story, perhaps slightly in the direction of your question/request.
Here is one survey article on Hilbert's 16th Problem, where this issue is on p 5, section 4, problem 3.