Not exactly illnesses, but in earlier medical history, illnesses were often associated with one of the four humours, the names for which have come to mean character traits - phlegmatic, sanguine, melancholic and choleric.

Other illness related words that have come to have a wider application would include: manic, poxy, cancer (-ous, etc), demented, dizzy, fever (febrile, etc), infectious, jaundice(d), splenetic, lousy, scab (scabby, etc), vertiginous.

There are also other words that only coincidentally mean other things, being applied to both, rather than having originated with an illness, such as bipolar, stroke, shingles, thrush and shock. Not sure which category measly, rash and impotent fit into.