It's hard to respond to stuff like this. I get the attitude: "I'm due what I paid for!"
There are so many ways in our society where there's unwritten expectations. In this case, the unwritten expectation was that they weren't going to meter you and you didn't have to worry about overage charges, but that the 1Gbps should be thought of more as a burst rate than a 24/7 sustained rate. Likewise, I'd interpret it to mean that if I had a heavy month, I'd be fine.
If I'm a student paying for a dorm in college, should I be allowed to set up a mining rig pulling 10 kW and costing around $720/mo to run? Electricity is included in my dorm fee! I'm paying for it and I'm going to use it!
This is kinda why AWS nickle-and-dimes customers for every little thing - they had people who would find any area of weakness in their pricing model and go after it. I'm not even saying that people did it maliciously. Jungle Disk was a really cool idea: a mountable disk that used S3 to store the files. The problem was that it started doing loads of S3 operations. At the time, AWS didn't charge for operations and now there's this popular program doing a lot of operations compared to its storage and transfer charges.
As someone that has created SaaS software, some customer always comes along and finds an interesting use of your software. Sometimes it's a customer not really understanding the product. Other times a customer has found a weakness in the pricing model and they want to exploit it.
I get it the attitude. It's a war between you the buyer and the company selling a product to you. Personally, I'd rather have a more amicable relationship in my dealings. Yes, that means having realistic expectations about offerings including what might be marketing "puffery" (whether it meets the legal definition of puffery or not; https://en.wikipedia.org/wiki/Puffery). I'm not going to say that you're wrong to want 100% of the link 24/7, but then you're just talking about being in a situation where the company exercises their right to terminate your account (which you would have agreed to when signing up). Looking at Hetzner a bit more, it looks like they charge €1 per TB or $0.001/GB on their plans that have overages - 10x better than cheap places like Digital Ocean.
Yea, there's lots of places in this world where you can cause headaches for other people by using something in a way it wasn't intended that you're entitled to do - but you're going to have an unhappy time fighting for what you're due all the time. Instead, I'd rather focus on using things the way they were intended such that I don't create adversarial relationships with those I work with. Yes, in this case it would mean that I'd probably buy a few servers or just offer to pay them €200 for the extra bandwidth. Or I could fight, they could give me 30-day notice, and I could find another "unlimited" provider until they see me as a toxic customer and I start it all over again. Again, you're not wrong. You're just going to have an unhappy time trying to enforce your rights and get something better than the company expected to be giving you.
And of all the scummy things out there, Hetzner seems to be pretty decent. €0.001/GB overage on their limited plans is really cheap even compared to good deals like Linode and Digital Ocean (which charge 10x that). Their servers are really cheap. Their pricing is mostly very straightforward - maybe with this being an exception. I'll take a nebulous "it's unlimited, but don't push your luck" over what most hosting providers are offering. I could probably do 250TB for a month and they'd ignore it if it wasn't consistent. I can transfer way more than I get Linode or Digital Ocean and they'd ignore it. Maybe you just want me to say that you're right and that Hetzner is being evil and the person is due their 250TB of transfer every month, but I'd rather go through the world thinking that things are run by humans who want to do the right thing and sometimes don't anticipate all the consequences and then something like this happens - especially when the company doesn't seem to be doing evil stuff. Like, we've all seen surprise AWS bills in the tens of thousands without enormous usage. In this case, we're talking about a pretty small bill compared to the usage.
Yep, that person has the right to use that bandwidth - and Hetzner has the right to terminate a customer for any reason with 30 days notice. I'd just rather try and work with a company in a reasonable fashion if they're willing to work with me in a reasonable fashion. The alternative is becoming the customer that the company hates.