After Voldemort took hold of the Ministry for Magic, a Muggle-Born Registration Commission was appointed. The purpose of this Commission was to interview all Muggle-borns (DH11).
Head of the commission was Dolores Umbridge (DH13). She interviewed the Muggle-borns in the presence of Dementors, together with Yaxley and Mafalda Hopkirk as scribe (DH12). The commission imprisoned all Muggle-borns they judged in unfair trials because they would have stolen their magic from wizards (Pm).
One of the Muggle-borns interrogated was Mary Cattermole. Harry, disguised as Runcorn, rescued her and other Muggle-borns and advised them to leave the country (DH13).
Ron presented this argument against the claim of the commission: "if you could steal magic, there wouldn't be any Squibs" (DH11).
Also, Rowling later stated that all Muggle-borns have a witch or wizard somewhere on their family tree (BLC).
Rowling had already devised the pure-blood / mudblood divide in the Harry Potter world, when she discovered that the Nazis had had the same idea: they stated that a single Jewish grandparent "polluted" the blood (JKR).