The last paragraph in the Google thread referred to in previous answer says: "I suppose most of what I just wrote is merely conjecture. Still, one thing we can be sure about is that "Yours, etc." has been in use since 1813, when Pride and Prejudice was published, and one can assume that it was used further back than that."
Indeed, it's used in one of the earliest English novels, Samuel Richardson's Pamela, or Virtue Rewarded. In 1739, at the request of friends who were booksellers, Richardson began writing Pamela as a "letter-writer", that is, as a volume of model letters for "country readers" "unable to indite for themselves". Of course, he ultimately turned the volume into a novel.
Phrases like "I am, &c.", "From, &c.", and "Yours, &c." appear at the ends of a handful of letters in Pamela, while dozens of times as many letters have valedictions like "Your most dutiful DAUGHTER.", "Your dutiful daughter,", "Your dutiful DAUGHTER till death.", "Your honest as well as dutiful DAUGHTER.", "Your ever dutiful DAUGHTER.", "Your most afflicted DAUGHTER.", "Your dutiful and honest daughter,", "Your careful, but loving Father and Mother,", "Your truly loving, but careful, FATHER and MOTHER.", and so forth.
In Pamela the "... &c" valedictions appear without explanation, they are relatively rare, and the correspondents are mostly Pamela and her parents, so it's difficult to draw real conclusions. It does seem clear that prolific correspondents would have little trouble interpreting "I am, &c." in accord with formula, but that is not to say they "will understand exactly what has been abridged" since there is much variability within the framework. I imagine that consulting a contemporaneous letter-writing manual is the only way to answer your question definitively.
 Introduction, W. M. Sale, in 1958 Norton Library edition of Pamela, or Virtue Rewarded