No. Dual band just means you have two options to transmit and receive a signal. But receivers will only ever use one band to connect with. Think of it like a choice of using two roads to reach a destination.
If we're talking wireless n, there are advantages to both bands. The 5Ghz band is faster and is a less crowded spectrum, given that it's relatively newer, so you will be less likely to suffer interference from your neighbours on the same channel. It also uses a broader spectrum, so is better for e.g. streaming HD movies. However, the 2.4Ghz band has greater range and is less susceptible to interference from e.g. walls obstructing the signal, so can offer greater stability in larger properties or ones with many walls.
Wireless ac only uses the 5Ghz band. It is backwards compatible so will connect to older, n equipment through the 2.4Ghz band where necessary.
If you are in the market for a new router dual band is a sensible choice. Most decent routers support it these days and it gives you much greater opportunity for a good connection.