Do companies have ways to know if an applicant has a degree if...?

they did not specify it on their application? Ex: Instead of saying they have a bachelors or masters degree, they circle "high-school diploma" in the "highest education completed" section. If they don't want to be considered overqualified, would they have to divulge such information? How would an a potential employer truly know of the highest education completed by an applicant if all they say is a high-school diploma? Can they verify that you have a degree, even if you state otherwise?