TITLE: Insights from disproofs after counterexamples have been given
QUESTION [6 upvotes]: Some conjectures are disproved by a single counter-example and garner little or no further interest or study, such as (to my knowledge) Euler's conjecture in number theory that at least $n$ $n^{th}$ powers are required to sum to an $n^{th}$ power, for $n>2$ (disproved by counter-example by L. J. Lander and T. R. Parkin in 1966).  But other conjectures retain interest, even after being disproved by counter-example, and lead to analytic disproofs and insights.  
Question
What are some conjectures, first known to be false through counter-example, but whose subsequent analytic disproofs shed novel insights into the problem?  More generally, what are the properties of known false conjectures that nevertheless garner significant interest and efforts of mathematicians to produce analytic disproofs?

REPLY [2 votes]: The first two things that come to my mind are actually the dual of what you asked for, namely conjectures where the counterexample did not come first, and the search for an explicit counterexample led to interesting developments.

Brosnan and Belkale disproved a conjecture of Kontsevich about polynomially countable graphs, but their argument did not lead to a specific counterexample.  An explicit counterexample was provided by Dzmitry Doryn.
That $\pi(x) > \mathrm{li}(x)$ for all $x$ was disproved by Littlewood, but without giving an explicit counterexample.  I believe that there is still no explicit counterexample, but the search for one has inspired some interesting work.