If I only had 3 days of sick for every 6 months, I would be inclined to come to work if I had a cold/minor issue too. I'd be thinking that I may need it for something real.
My employer also encourages sick people coming to work in that after you accrue a certain number of sick hours it converts to vacation at the beginning of the year. So if you are maxed out on sick leave you earn an extra 2 weeks of vacation per year.