Aggressive geometry – Non-buildable conditions on fibers that pass from the generic point to a non-empty opening

Let $ f: X rightarrow Y $ to be a flat morphism of schemas, with a locally irreducible Noetherian target. Call a condition on the fibers of $ f $ "good" if the condition is true in the generic point of $ Y $ if it is valid on an open non-empty set. What are some geometrically interesting examples of "good" non-buildable conditions on fibers?