Perhaps one of the most famous computational model discoveries of the 20th century is Lorenz's observation in 1961 of chaotic behavior in a weather model. It overturned the existing consensus in meteorology, and led him to the discovery of a strange attractor in a simplified model of atmospheric convection. The implications are described in the now famous 1963 paper Deterministic Nonperiodic Flow, that started modern chaos theory, where he stated "In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent." In a 1966 paper Arnold gave a general theorem about a class of models that confirmed Lorenz's insight, in his words he "used it to show that weather prediction is impossible for periods longer than two weeks."
Wikipedia describes the circumstances of Lorenz's discovery:
Lorenz was using a simple digital computer, a Royal McBee LGP-30, to run his weather simulation... To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 was printed as 0.506. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However, Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.
But I have to quibble a little with the idea that an experiment can "disprove" a theory. Modern theories are too sophisticated to be disproved by a single experiment, or even by a group of them. Experimental data is based not only on a theory but also on a number of extra assumptions (coming from multiple other theories and "common sense") that can be changed or discarded leaving the theory intact. The Michelson-Morley experiment for example was quickly reconciled with the ether theory by Lorentz, who proposed that (contrary to "common sense") rulers contract and clocks slow down when moving through ether. In fact, Michelson himself still believed in 1902 that the ether theory was triumphing, which was the consensus at the time. Developing a scientific theory is a complicated process, that involves experiments, theoretical analysis, numerical simulations (or just computations and estimates by hand), etc., and any of these stages can provide what is later seen as "the critical step" in disproving a theory.