[This is a fast-access FAQ excerpt.]
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.