Time and place are hazy--at various times while I was in school, I worked on projects involving various Departments of Health--you may remember back to the days of Model Cities and the War on Poverty--somewhere, in a discussion of community health issues, someone was talking about having to dignose illnesses from based on second or third person accounts, and giving examples, one of which was, you know it's diphtheria if someone mentions a bubble in the throat--at which time I recited the jump rope rhyme, and the person said, "Hey, that's about a child dying of diphtheria."

You don't have to go back to my source, though--look up Diphtheria--it was a serious health problem up through the 1940's--up to 150,000 cases a year(in a much,much, smaller America)--with a high risk of fatality, particularly for small children.

I don't know how or where the rhyme originated--I am familiar with a lot of the other related rhymes as well(the "naughty ones", I particularly recall, for some reason;-), but my version is as I learned it--it seems unlikely to me that the line "died last night with a bubble in his throat" could refer to anything other than diphtheria, though it certainly could have been interpolated into another rhyme--