Usually yes — DayQuil contains acetaminophen (not an NSAID), so adding ibuprofen doesn't double up on the same drug, and acetaminophen and ibuprofen can be taken together — but follow the dosing limits and check with your pharmacist first.
Per its FDA/DailyMed label, Vicks DayQuil Cold and Flu contains acetaminophen 325 mg, the cough suppressant dextromethorphan 10 mg, and the decongestant phenylephrine 5 mg per 15 mL — it does not contain an NSAID. Ibuprofen is an NSAID, a different class of drug from acetaminophen, and the NHS says adults can take ibuprofen at the same time as acetaminophen (paracetamol) if needed. The main hazard with DayQuil isn't ibuprofen itself but accidentally taking a second acetaminophen product: DayQuil's label warns "do not use with any other drug containing acetaminophen" because too much can cause severe liver damage — and ibuprofen does not contain acetaminophen, so it doesn't add to that load. The added caution is that ibuprofen brings its own NSAID risks (stomach and kidney), and DayQuil's phenylephrine has its own cautions in people with high blood pressure or heart disease.
Stay within each product's limits: DayQuil's label caps you at 4 doses in 24 hours (this is your acetaminophen ceiling), and the NHS limits over-the-counter ibuprofen to no more than 1200 mg/day (e.g., 400 mg up to 3 times daily, at least 4 hours apart). Before taking either product, do not pair DayQuil with any other cold/flu remedy, Tylenol, or combination pill that also contains acetaminophen — that is the real overdose risk. Ask a doctor or pharmacist first if you have a stomach ulcer or a history of GI bleeding, kidney/heart/liver problems, asthma or allergies, or are pregnant or trying to conceive (ibuprofen may be unsuitable), or if you have high blood pressure, heart disease, thyroid disease, diabetes, or an enlarged prostate (cautions tied to DayQuil's phenylephrine decongestant), or if you take an anticoagulant such as warfarin or steroid medicine (the NHS lists these among medicines that may not mix well with ibuprofen). Watch for stomach pain or indigestion, and stop and seek urgent care for signs of GI bleeding (vomiting blood, or black, tarry stools). Avoid alcohol, and call a professional if symptoms last more than a few days or if you feel you need both around the clock.
This is general reference, not medical advice, and not a guarantee of safety. Interactions depend on your doses, health conditions, and other medicines. Always confirm with your pharmacist or doctor before combining products, and follow the dosing on each label.