"Mam czerwony nos."

Translation: I have a red nose.

[dfwgator]

Rudolph?

[HubertTybus]

If someone has "a red nose", it may means he drinks much alcohol.

[Euhan1]

I've heard of this happening once in India.

[Helena834099]

Or he was too long in the African sun.

[Ollyfer]

it could be worse if he or she was a regular drunkard—then, the nose may be red, swollen and veined. Maybe there is a word for it in Polish too; in English, the only translation offered to me simply was “a drunkard's nose”.

[SaraWie]

Is 'czerwony nos' accusative? Do you always use accusative when using 'I have a /body part related thing/?'

[Jellei]

You simply use Accusative after "mieć" (to have) :) Yes, it is.