Does a router know not to hand out static IP addresses to other devices?

"Your router can indeed lease that IP address to another client once the lease expires if your machine is off or disconnected, and doesn't renew it. To guarantee you always get the same IP address, you'll want to make what's called a DHCP Reservation in your router (terminology may vary - see this question for details):

With DD-WRT firmware, you can find this under the Services section:"

"Agreed with the above. You can either setup a DHCP lease reservation for the addresses in question so that the DHCP server always gives that address to the machines in question, or you can define a static IP address that is outside of the DHCP pool.

I prefer option #1, using leave reservations. You handle all of your addresses in one place and are prevented from assigning the same address to more than one device. Also, for most devices with static IP addresses, they will not show up in the list of devices connected to the network due to most routers pulling that information from the DHCP lease information in the router. Client-side static IP addresses are very easy to mess up unless you keep a diagram or spreadsheet to keep them straight."