You can use the phrase and it would be understood. However, falling sick is usually considered an event - it either happens or it doesn't. More often you would hear that a person fell sick and became slowly worse or sicker. You can say "fall very ill", because it can still refer to a single event.

I agree that it's based on context. But "Slowly fall sick" isn't incorrect, just sounds unnatural to me because I can't think of a context in which I would say it. I do however think it sounds fine in past or future tense: "During the week I slowly fell ill and it ruined my weekend since I had the flu." "If he doesn't take his medicine, he will slowly fall ill and then die." Just some of the perks of speaking a language with no governing body and 100s of variants and dialects.

slowly falling ill I think it would be based on the rest of the sentence

You'll hear in formal stuff (like old books) or in certain social settings (often older people) say something like this : "I appear to be coming down with something." Which is an expression to mean : I am slowly becoming sick. What you will probably end up hearing is people say "I'm getting sick" or "I think I'm getting sick" or "I think I'm getting the flu" or "I think I'm getting a cold." And when they say "getting" it will sound like "get in." However, don't let that confuse you there are just saying getting, but shortening it in their speech. Why? Are you getting sick? One more thing, all people tend to use coming down, when talking about children and fevers. Mothers will put their hand on a child's forehead, and say "I think you're coming down with something" and then "I think you're getting a fever."