The answer is no. But to explain why I want to not talk about the human immunodeficiency virus, which we now know causes AIDS (the acquired immune deficiency syndrome), but by TMV -- the tobacco mosaic virus. The first true "germ" was isolated by Koch in 1876 -- anthrax, quickly followed by TB and cholera. The impact of germ this new germ theory (codified in Koch's postulates) on medicine cannot be underestimated. And in Europe, within a decade of its discovery, germ theory had been whole-hardheartedly embraced (it took another decade in the US for it to fully catch on) -- so much so that by the 1890s, a new dogma had taken hold that all "viruses" (which previously referred to any disease-inciting agent) were caused by bacteria. Pasteur himself wrote "tout virus est un microbe." Scientists were discovering all sorts of microbes by using filters with small pores to isolate them, then taking the isolates and introducing new disease in laboratory animals.

Which brings us to the tobacco mosaic contagion. This disease, which caused "mosaic" destruction in tobacco plants, had been recognized throughout the 19th century, and it had been noted that it had been spread by touch. Prior to germ theory, the working assumption had been that the disease was spread by an "enzyme", that is, some sort of chemical reaction. But with the advent of Koch's postulates, scientists started to think the disease was caused by a microbe. Despite this, no one was able to isolate a microorganism using Koch's postulates from diseased tobacco. Ivanovsky performed the first famous experiment trying to "filter" out a bacterium, and when he could not, he hypothesized that there might be bacterium so small that we could not see them. A few years later -- 1898 -- Beijerinck performed his famous experiments on the tobacco mosiac contagion and determined that the disease was NOT caused by a microbe, but by a "non-cellular" entity, which he called a contagium vivum fluidum, also referred to as a virus (a separate entity from the old definition).

Over the next few decades, there were more experiments elucidating more enigmatic "viruses" in animals, including humans. But it wasn't until 1935 that doctors at Princton were actually able to isolate the first virus -- of course, the famous TMV.

My point here being, scientists prior to the 1898 had NO concept of viruses, and prior to 1876 would have had no concept of infectious disease as we now know it. Even if they had come across someone with AIDS (so someone who presented with, say, Kaposi's sarcoma and a terrible pneumonia like PCP), they wouldn't have had the intellectual frame to define the disease. Though apparently Kaposi himself didn't describe his sarcoma until the 1870s.

So how far back does AIDS go? AFAIK, the first suspected case is on Robert Rayford, a teenager from Missouri who died in the 1960s. As for HIV, the oldest known HIV infection is from a lymph node from a patient from modern day Kinshasa (then Leopoldville), from 1959. The nature article above suggests that genetic distance between this HIV-1 and modern HIV-1 genomes suggest that HIV at least had a considerable time to evolve (though it's unclear if this strain would have caused AIDS).

I hope that sort of answers your question -- while it seems likely that HIV existed before the 20th century (whether or not AIDS existed is an open question), medical science was far from ready to have recognized it as its own entity.