It depends a lot on the particular wadcutter. A substantial portion of a projectile's stability derives from its geometric and gravimetric uniformity. A combination of the rifling spin and the pellet's shape helps to counteract the real-world imperfections in these properties. But if we start with a pellet that is supremely uniform, it will remain stable for a longer distance.
We usually talk about things in terms of accuracy so we should probably draw a distinction between stability and accuracy/precision. A loss of stability produces things like spiraling and keyholing, which certainly diminishes accuracy in a severe way. However the vastly poorer BC of a wadcutter will always produce larger groups at a distance than a similarly quality dome because of wind influence, even if the wadcutter is still objectively stable at that distance.
For example I have a batch of 9.3gr RWS Supermag wadcutters that are just excellent from almost every .177 barrel I own. On days with low wind, they will bughole at 30 yards:
https://www.youtube.com/watch?v=khiYSopVXes
They are still producing nice clean holes in paper on my 50 yard target, indicating they are still stable at that distance. I've even had some good groups at 50 but they are rare and have never equaled my best groups with domes.