English Experts

[Marcio_Farias]

[point taken]"Come down sick" pode ser usado por nativos.

donay, dando uma olhada no dicionário[1] encontrei também "take down" usado com "sick":

take down
intransitive verb
1 : to become seized or attacked especially by illness <took down with typhoid fever> <man ... who was bearing most of the fitting-out expense took down sick — J.F.Dobie>

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[1] Merriam-Webster's Third New International Dictionary em CD v.3.0.

[Marcio_Farias]

Still according to my copy of Webster's, sick in that case functions as a "subjective complement" (a grammatical complement relating to the subject of an intransitive verb <in "he had fallen sick" sick is a subjective complement>)

[rotahc]

Also...

to be taken ill