Yes, that's exactly what happened! It was actually fairly early in the history of Sodom, shortly before the war of the 4 and 5 kings.
The people of Sodom murdered poor Humpty because they knew he was wealthy and wanted to steal his property. Their plan backfired: the murder became famous, and the 4 kings heard about it and decided to come and take that property for themselves.
The "king" in the song is none other than Avraham, who became king at the time of that war. Although he said to the king of Sodom that he wouldn't accept any property, of course he would have returned Humpty's property if he could have, and so he tried to put him together again.
The girsa of the song is actually corrupt. Avraham is never mentioned as having any horses, and there probably weren't any horses in Israel yet. The correct and original girsa, from R' Shmuel Arnold, reads:
Humpty Dumpty sat on a wall,
Humpty Dumpty had a great fall.
Four-score Men and Four-score more,
Could not make Humpty Dumpty where he was before.
It says "four-score" twice, which means 160 men. "and" and "more" are both ribbuyim that add an additional four-score each, making 320. Finally, miut "men" shnayim, giving 322. That's how many men Avraham had: 318 as mentioned in the pasuk, plus Eliezer, Aner, Eshkol, and Mamrei.
Unfortunately, they weren't able to put Humpty back together. Avraham gave the property to his heirs, the Dumpty family, who weren't living in Sodom, rather than to Humpty's murderers. That's why it says וישב את כל הרכוש but not את כל העם: he gave all the wealth back to its rightful owners, but couldn't return all the nation because Humpty was missing.
If Humpty had still been living in Sodom, maybe he would have been enough to boost the total over 10 tzaddikim. But they killed him and the city had to be destroyed.