A dealer might offer a fixed price for the car you are buying but give you a low-ball offer on your trade-in. Always get a separate quote for your current vehicle from other sources first.
If you want a no-haggle experience at a traditional dealer, email the Internet Sales Manager at multiple locations to request their best "out-the-door" price before you visit the showroom. No Haggle Car Pricing Buying Guide - Consumer Reports
Dealers like CarMax or those using Consumer Reports Build & Buy set prices that are often competitive but may not always be the absolute lowest possible market price.
No-haggle doesn't mean "no-profit" on the loan. Check your bank's pre-approval rates before arriving to ensure the dealer isn't marking up the interest rate.