'immune-incompetent' is a compound of 'immune' + 'incompetent'. 'immune' originates from Latin 'immunis', where 'im-' (negation/exemption) + 'munis' meant 'obliged/charged' (so 'exempt'); 'incompetent' originates from Latin 'incompetens', where 'in-' meant 'not' and 'competens' meant 'suitable/fit'.

'immune' passed into English via Late Latin and Old French (Latin 'immunis' → Old French → Middle English 'immune'); 'incompetent' came from Latin 'incompetens' via Middle French/Medieval Latin into Middle/Modern English. The compound 'immune-incompetent' is a modern English medical formation combining the two elements.

Initially 'immune' could mean 'exempt (from public service/tax)'; over time it shifted to the sense 'protected from disease'; 'incompetent' originally meant 'not meeting standards' and has retained that core sense, now applied in medical contexts to describe inadequate immune function in 'immune-incompetent'.