How long should I keep waiting before giving up on refinancing, having bought at 6.625% with 3% down in 2023?

There's no give-up date. The break-even math tells you when a refinance is worth doing, and until then you keep the option open. Nobody can promise where rates go from here, and that cuts both ways. Rather than staking everything on one magic rate that may never arrive, check the numbers periodically and act when they work. Worth knowing: a refinance doesn't have to reset you to a fresh 30 years. After a couple of years of payments, the new loan can be structured to match the term you have left, say a 28- or 25-year, so you keep the progress you've made while still capturing a lower rate or payment. Custom terms like that mean there's almost always a structure that helps once rates cooperate. The test each time is simple: what does the refinance cost, and how many months of payment savings does it take to earn that back? A reasonable window says go; anything longer says wait and look again. A good loan officer can also put you on a rate watch with a realistic target so you're notified when your number hits instead of guessing. If you want us to run your specific balance and set that threshold, that's exactly what the free Roadmap conversation (about 20 minutes) is for: we run your real numbers.