I bought a home in June with a 4.75% rate, 30-year fixed. I now see that many places are advertising rates around 3.8%. Assuming I can get a low or no fee deal, would it be silly to try and refinance less than 6 months after a purchase?
The advertised rates are usually for 20% down (or refinances with 20% equity). I'm guessing that having bought so recently, you don't have that much equity? Give the banks advertising those rates a call and check what the requirements (equity and credit score) are for those loans; that will probably answer your question.
Also, most lenders won't refinance within six months of a purchase, and some require a year. They call this "seasoning" (they won't refi until the initial loan is "seasoned").