This article is from the alt.usage.english FAQ, by Mark Israel misrael@scripps.edu with numerous contributions by others.
Yes, this means "Don't let the bastards grind you down", but it
is not real Latin; it is a pseudo-Latin joke.
"Carborundum" is a trademark for a very hard substance composed
of silicon carbide, used in grinding. (The name "Carborundum" is a
blend of "carbon" and "corundum". "Corundum" denotes aluminium
oxide, and comes to English from Tamil "kuruntam"; it is related to
Sanskrit "kuruvinda" = "ruby".) "The "-ndum" ending suggests the
Latin gerundive, which is used to express desirability of the
activity denoted by the verb, as in "Nil desperandum" = "nothing to
be despaired of"; "addendum" = "(thing) fit to be added";
"corrigendum" = "(thing) fit to be corrected"; and the name Amanda,
from "amanda" = "fit to be loved").
"Illegitimis" is the dative plural of "illegitimus" =
"illegitimate"; the gerundive in Latin correctly takes the dative to
denote the agent. "Illegitimus" could conceivably mean "bastard" in
Latin, but was not the usual word for it: "Follett World-Wide Latin
Dictionary" (Follett, 1967) gives "nothus homo" for bastard of known
father, and "spurius" for bastard of unknown father.
The phrase seems to have originated with British army
intelligence early in World War II. It was popularized when U.S.
general Joseph W. "Vinegar Joe" Stilwell (1883-1946) adopted it as
his motto. Various variant forms are in circulation.
 
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