It depends entirely on which Mucinex: plain Mucinex (guaifenesin only) doesn't share any ingredient with DayQuil and is generally compatible, but the many combination Mucinex products (DM, Cold & Flu, Fast-Max, Sinus-Max) duplicate DayQuil's acetaminophen, cough suppressant, or decongestant and can cause an overdose — so read both labels and ask a pharmacist before combining.
Regular Mucinex contains only guaifenesin, an expectorant that thins mucus, while DayQuil (Cold & Flu) contains acetaminophen, the cough suppressant dextromethorphan, and a decongestant (phenylephrine) — different ingredients that work in different ways, so adding plain guaifenesin to DayQuil does not duplicate anything. The danger is that "Mucinex" is a brand with many versions: Mucinex DM adds dextromethorphan (same cough suppressant as DayQuil), Mucinex D adds a decongestant, and Mucinex Fast-Max/Cold & Flu/Sinus-Max versions contain acetaminophen plus dextromethorphan and a decongestant — the very same ingredients as DayQuil. MedlinePlus warns that nonprescription cough and cold products "may contain the same active ingredient(s) and taking them together could cause you to receive an overdose," and the FDA DayQuil label warns of "severe liver damage" and says do not use with any other drug containing acetaminophen. Doubling up on acetaminophen (liver harm), dextromethorphan, or a decongestant is the real, label-stated risk here.
First, identify your exact products: if it is plain Mucinex (guaifenesin only) plus regular DayQuil, there is no ingredient overlap, but still follow each label's dosing and do not exceed the DayQuil acetaminophen limit (max 4 doses/24 hours; severe liver damage can occur above that, and the risk rises with 3+ alcoholic drinks a day). Do NOT combine DayQuil with a Mucinex product that also lists acetaminophen, dextromethorphan, or a decongestant (phenylephrine/pseudoephedrine) — that means most Fast-Max, Cold & Flu, Severe, DM, and D versions — because you would double-dose. Watch for signs of acetaminophen overdose (nausea, vomiting, upper-right abdominal pain, which can appear even before you feel sick — seek care immediately and call Poison Control at 1-800-222-1222) and decongestant overlap (jitteriness, racing heart, raised blood pressure). People with liver disease, high blood pressure or heart disease, those taking an MAOI antidepressant, who drink alcohol heavily, or who are giving these to a child should not self-combine — ask a pharmacist or doctor first. When in doubt, take one multi-symptom product rather than stacking two.
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.