Cases mentioned below where space fills up quickly due to a bug, maybe yes. Outside that there's the problem that you can ignore the emails (or be sick, asleep, etc). Worse if they go to a team and everyone is busy and assumes someone else will deal. Bad if you aren't in charge and tell people in charge and they nod and don't decide anything - they prefer to run at 70/80/90/95% used indefinitely instead of signing a cheque.
When the drive fills and everything breaks and, you /have/ to respond, and it becomes socially ok to make it your highest priority and drop everything. An email "only a few weeks until maybe it runs out of space" is much harder to prioritise and get anyone to care. With this system, the time when it fills and breaks has some flex for you to not go down with the server, and save your own bacon. It's as much a fix for the organization as anything else.
I see this most in smaller company aging systems where they had ample storage and drives for a few years ago when they were new, now they're crammed full with the growth of software, libraries, updates, data, new services being deployed on them, increased demands, and nobody wants to commit to more storage for an older system towards the end of its warranties, but they definitely don't want to commit to the much larger cost of a replacement and all the project work, and running at 90% full costs nothing and involves no decisions. 91%. 92%.