If not, then how long? (Assuming with infinite supply)
After a certain time people would get nutrient deficiencies (vitamins and minerals). When deficiencies start to occur depends a bit on each individual person. We have Vit's E, D, K and A as fat soluble vitamins stored for a while. Vit C, B's and others would diminished sooner if not replenished. You would also lack minerals such as calcium, magnesium, potassium and so on. So you will get sick at some point.