It would internally black out large swatches of the Internet
Sure. Let's say they do the common thing of using private IP addresses internally to their network, such as 10.x.x.x... You know the drill, network address translation at the edge of their network, just like your home network.
Except they decided 10.x.x.x is too restrictive for them, and they start assigning public IP addresses internally. It will work, at first. But then problems will start popping up.
It's a matter of time before somebody uses 188.8.131.52 for a lab machine. It's one of the IP addresses DNS resolves for www.google.com. Now, sometimes, when someone inside the university tries to do a search on Google, their web browser goes to that lab machine instead. Because the internal routers would have no ability to conceive that there are two 184.108.40.206's, one internal and one external; they would simply route your packets to the internal one.
IP blocks assigned internally cannot be routed externally
But it's worse than that. They assigned a whole netblock, so all of 172.217.x.x/16 will route to that lab. You probably wouldn't clobber every Google IP, but a lot of searches would fail. For smaller outfits like Craigslist where all their addresses are in the same netblock, if the university assigned that netblock internally, the entire site would be blocked cold.
This won't affect anyone outside the university's internal network. External providers will not accept the university's reassignment of Google's IP space. The only traffic routed to the university will be the public IP addresses that the university owns.
Just use IPv6 instead
If you sign up for Comcast, they give you a /64 of your very own. If you ask nicely, I've heard they'll just hand you a /48. But let's say you only get a /64, and then, do exactly the plot of RevOlution and create self-replicating nanites that eat electricity, in the same quantity as discussed on the show. Do you have enough IPv6 addresses for every nanite to have its own?
Yes. And enough spares to do this on 2 million parallel earths.
So if you're really worried about running out of IP addresses, that is the way to go.