That's certainly the oft-repeated modern belief, but there seems to be no evidence to support it.  According to Peter and Iona Opie (The Oxford Book of Nursery Rhymes, 1951) no examples of the rhyme are known in England prior to 1881, though a version sung to the familiar tune (but with no sneezing or falling down) was reported from Massachusetts c.1790.

In fact, sneezing and falling seems to be an introduction of the late 19th century, which has only become widespread during the 20th; the Opies comment, "The foreign and 19th century versions seem to show that the fall was originally a curtsey or other gracious bending movement of a dramatic singing-game."  They go on to quote analogues from Germany, France, Switzerland and Ireland (in both English and Gaelic).