Can two different Minecraft accounts play on LAN together when both having the same IP address?

My little sister just brought me minecraft so i can play on LAN multiplayer with her. We're at the same house, so my laptop and the computer have the same IP address. Every time we set up the LAN server it won't let me connect and says " java.net.ConnectException: Connection refused: no further information: "
and both of us can join other multiplayer games on LAN
does anyone have a solution to our problem?

Added (1). Also we both have separate accounts, in case I didn't add that in there.

You may both have the same public IP address on the other side of your internet router, but you can't have the same IP address on the LAN side of your router or neither of you would be able to do much of anything on the network (IP conflict). It is possible that a firewall on the computer you are trying to connect to is blocking. But one issue with "Open to LAN" is that it uses a different random port every time it runs, so just opening the usually minecraft port 25565 will not help (unless running a regular minecraft server instead of Open to LAN).