“Why look for a counterexample if the conjecture is so obviously true? Well, because you always should.”
The bunkbed conjecture sat open since the early 1980s because everyone was sure it was true. The counterexample is a graph with 7,523 vertices and 15,654 edges. The probability gap is on the order of 10 to the minus 6,500, far too small to check by brute force, so the disproof is a 12-page combinatorial argument. Two of the three authors are graduate students. Obvious is not a proof.