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RFC 1918 specifies 3 ranges of IP addresses as private
10.0.0.0/8
172.16.0.0/12
192.168.0.0/16
So in a real network, only these IP addresses could be used? What if I assigned a host on the network an IP of 12.1.1.1, for example. Would there be a problem?
You could, but you shouldn't. If you do so, the real devices and services using that IP range become unreachable.
I've seen a number of customers make this mistake (sometimes intentional, sometimes not), and it always ends up with renumbering the internal network because the public IP's used internally needed to be accessed.
Indeed you can but you shouldn't. What happened in the past, at the beginning of the Internet, was that companies rented public C classes from IANA because it was cheap ('90s/2000's)and used that to build internal network; then they stopped paying for such public IP addresses but didn't spend time to renumbering it; the result was what @teun said, i.e. that it was not possible for them to reach the public IPs used internally. And yes renumbering a network is a pain in the neck...
