After reading a bit on the internet, I think I've figured out the answer to my question.
Strunk and White's style guide says that "When the subject is the same for both clauses and is expressed only once, a comma is useful if the connective is but. When the connective is and, the comma should be omitted if the relation between the two statements is close or immediate.
The case mentioned in the original question was the following:
"I don't know a fig about my sickness, and am not sure what it is that hurts me"
In this case, the subject is the same for both clauses and is expressed only once, and the relation between the two statements is not close or immediate. The second statement is distinct and emphasizes a slightly different point. Therefore the comma makes stylistical sense.
If, on the other hand, the sentence had been closer to "He has years' experience and is thoroughly competent", it would have been grammatically correct to omit the comma since the two statements are very closely related.
Let me know if this makes sense, and if I'm interpreting it correctly.