George Silver (Paradoxes of Defense, 1599) makes an appeal

... to reduce the wearing of swords with hilts over the hands, to the Roman discipline, no longer then they might draw them under their arms, or over their shoulders.

So drawing the weapon over the shoulder was a thing in Elizabethan England, at any rate. However, it must be pointed out that Silver's entire thesis was that the rapiers of Elizabethan times had become utterly absurd in length, to the point of mockery.

For, you honor well knows, that when the battle is joined, there is no room for them to draw their bird-spits, and when they have them, what can they do with them? Can they pierce his corslet with the point? Can they unlace his helmet, unbuckle his armor, hew asunder their pikes with a Stocata, a Reversa, a Dritta, a Stramason or other such tempestuous terms? No, these toys are fit for children, not for men, for straggling boys of the camp, to murder poultry, not for men of honor to try the battle with their foes.

He goes on and on about the sheer stupidity (as he saw it) of these Italian weapons and how unfit they were for strong English men. He was both right and wrong; wrong because the Italian systems ultimately prevailed throughout Europe and drove older more medieval martial arts to extinction. But right because they had grown absurdly long c. 1599, before the new science of fencing had caught up to the technology. Eventually the science did catch up, and rapiers underwent a serious shortening in the mid- to late-17th Century, eventually evolving into the small sword.

But we can infer from Silver's mockery that the over-the-shoulder sword draw was understood as a bit ridiculous in its time. This is especially interesting when you consider that it was contemporaneous with one of the largest of swords, the European two-handed sword or zweihander, a sword that should be carried over the back, if there ever was one. But virtually all illustrations of these weapons show them being carried over the shoulder, often with a normal-sized sword in a scabbard at the hip.

So the answer seems to be that the scabbard on the back did happen, but it was not nearly as pervasive as you might think from playing fantasy video games. If a sword was too long to wear at the hip, it was generally perceived as too long to be a personal sidearm or backup weapon, and if it was a primary weapon for use in battle, then it didn't belong in a scabbard, did it? It should either be in your hands, or stored safely back in the armoury.