Hi to all who will read this post.
I am not sure that my own US Census/genealogy story counts exactly as "invented relatives" (although it seems to involve persons who were not relatives of mine who were counted in the US Census as (in fact) relatives of a relative of mine) -- and may not involve persons who never existed, but (echoing Deanna Levinsky's noticing boarders being possibly "passed off as family") I have what seems to be a story of landlords (or possibly boarders -- in any case (if they actually existed) older persons) being "passed off as family".
In the 1910 US Census listing (from Brooklyn) for the household (on Johnson Avenue) in Brooklyn for my maternal grandmother's brother (who was then listed with his wife -- and no children; their ages are given as "23" (my great-uncle) and "20" (his wife) ) , a man (age given as "45") whose name is given as "Isidor" (possibly "Isidore") and a woman (age given as "43") whose name is given as "Annie" are listed as "Father" and "Mother" -- and all 4 listed persons are given with the family name of "Pate".
However, the 2 middle-aged persons (if they existed -- and whoever they were) could not have possibly been the parents of my great-uncle Hyman... -- as his sister who was my grandmother (the baby among her siblings -- actually younger than one of her nieces (!) ) was born on Madison Street on the Lower East Side of Manhattan (definitely not Madison Avenue) in May of 1910 (after the Census found her parents' household), and the household into which she was born (which naturally included both her and Hyman's parents) was duly listed in the same 1910 census on Madison Street in Manhattan -- with the parents' names given as "Isaac" and "Annie" Pat (with ages given as (respectively) "50" and "48" (Yes, it seems odd for my great-grandmother to have given birth in her late 40s, but that's "another story" (and her first children were born in the 1880s) ).
My great-grandparents (and their other children) subsequently took the name "Pate" that Hyman had come up with (and my grandmother actually didn't seem to remember having had any other family name when I talked to her about the "original" family name (around 1980, I think: "Pate" seemed unusual, and possibly changed (as it was...) ) -- but it seems clear to me that (whatever the situation was on Johnson Avenue in Brooklyn where Hyman and his young wife lived in 1910) the enumerator was somehow deceived (for some reason) into believing that Hyman's parents were living with him and his wife in Brooklyn, although they were alive and well (with 4 children -- 1 full-grown (age given as "22") and working living with them -- and another (my Grandma) "on the way") on a different island within New York City -- in Manhattan (on Madison Street, in what was then the Lower East Side).
I hope that this story is of interest and/or help in this thread.
Ethan Kent (who grew up on Long Island, and is now in New York City where he was born)
(Researching 4 main family lines: the Paat/Patt/Pat/Pates who emigrated from Bialystok (in today's Poland), the Kornhausers (who emigrated from Galicia -- from places in today's Ukraine (Turka) -- and probably today's Poland (Stefkowa)) , the Kantors (who emigrated from the Podolia region in today's Ukraine; possibly from Bratslav/Bratzlav), and the Gelperins/Halperins (who emigrated from the city known to Jews of their generation as "Vilna" -- but which is today known on maps as "Vilnius", the capital of independent Lithuania).)