Conventionally it seems accepted that to refute a theory requires only one counter-example. So I wonder, is it possible to have less than one counter-example, but more than none?
Do you mean theorem instead of theory? A theory doesn't claim to be the "whole truth" and when a counter example is found, it often just means the theory needs to be refined a bit. For example, the ancients knew that the heavenly bodies orbited, but they believed that the Earth was at the center of the universe. That made orbits really complicated with circles within circles to explain things like retrograde motion. Copernicus refined that idea and put the sun at the center of the solar system with everything orbiting in perfect circles. This worked OK, but eventually planets were not where they were predicted to be. Instead of completely refuting the old theory, Kepler changed the circular orbits to eliptical orbits and predictions became highly accurate. The only problem was that the predictions for Mercury were wrong. That did not refute Kepler's theory, but instead it had to be refined further by adding relativity to the picture.
Though I suppose with a theorem it's not a question of whether or not there is one counter-example, rather if there is any quantity larger than 0 counter-examples. Typically one is enough, but even a partial is likely sufficient to disprove it.
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