To come down with something is to start suffering from an illness. Come down with flu, come down with a cold, come down with something nasty.
It is used of the ordinary infections rather than of serious disease, so nobody comes down with cancer. The down suggests the illness arriving from outside and settling on you.
1phrasecome down withTo start suffering from an illness.
“Half the office will come down with it before the week is out.”
come down with is a phrase.
Also accepted in games: catch
Practised in Phrasal Verbs