You don't appear to be as aware of the networking as you think. But let's go anyway.
First, nothing stops you from assigning
111.222.xxx.xxx as your PC IP address. It's your PC after all, you can put anything you want. But you will accomplish nothing with this.
Why? A lot of reasons:
You probably aren't on the
111.222.xxx.xxx network (let's assume is
222.123.yyy.yyy. You must send the packets to the default gateway for your invented network, but your real network does not know it. You can try to add this as a default network, you OS will deny it, or put the rule and send packets nowhere.
Lets say you somehow managed to send the packets to your real default gateway, with an IP from another network. They will be dropped right away by the egress filter:
TCP/IP packets that are being sent out of the internal network are examined via a router, firewall, or similar edge device. Packets that do not meet security policies are not allowed to leave - they are denied "egress".
Your ISP knows all the IP blocks belonging to him. If any packet tries to get out of their network having a foreign source address, they will drop it.
If you got a badly configured ISP, and all the way badly configured upstream ISPs until the destination (absolutely, astronomically improbable), you will reach some endpoint somewhere. The OS will read the SYN flag (assuming it's a TCP packet), and send back the SYN+ACK. And you know where the packet goes? To the
111.222.xxx.xxx network, not your own network. It will never ever reach your PC. It will travel all way down to the rightfully owner of the IP you put on your PC, knock on a closed door, and be greated with an ICMP code 3 back:
Destination port unreachable, or will be silently dropped. And the connection attempt dies right there.
So, the answer is No. Nobody can put your IP address on their systems and frame you. But they can break into your system, install a proxy, and use a proxy running on your PC to do anything, and your IP will be all over there. Or they can break your wifi password, and do the same.