It's not uncommon to see people age anywhere from 8 to 14 years between 10-year census enumerations. If Adam was born in late 1880 or 1881, we won't find him in the 1880 census. But first, let's act thinking his age is accurate, that he was born in 1879. I'd check all the Kaiser census entry for the state of Illinois in 1880, looking for a 1 year old Adam. As Kaiser is a common name, I might search the census entries of William, Robert, and Adam Kaisers first.
Common surnames are troublesome.
You often have to keep records of a number of possible candidates and spend time just trying to eliminate candidates from your list. The Internet is filled with distant unknown cousins. The farther back in time you can identify your ancestors, the more likely another descendant of this ancestor has researched this individual and may have additional information on him or her. Take advantage of this information. Contribute your own information to the geneal genealogical database by publishing your findings.