Perhaps it is the portion of the hospital designed and equipped to handle Covid patients that is reaching a limit? To take my friend for example, he was at one hospital but on a list to be transferred to a hospital that had specialized equipment, in this case something called an ECMO machine that oxygenates blood. It was a 5 day wait until a spot opened.
It can be a thing, but a lot of times it's much more complicated (surprise!) than what a media headline reports. In your friend's case, ECMO is usually done at a CVICU (cardiac). Not all hospitals, even metro-area hospitals, have that capability (or capacity may be limited because it's not a specialty). So it's possible in your friend's case, it wasn't that there weren't any "COVID wards" available because "capacity," but instead, just normal CVICU availability. I get the end result may be the same, but it's not just because the hospital is "at COVID capacity."
Another thing that can happen is physical space. There may be plenty of capacity, but managing patient space has to be updated. There was a stretch in late 2020 and/or early 2021 where my wife's Peds CVICU unit got moved over to the PICU because they didn't have a lot of patients. They then moved adult COVID patients into the Peds CVICU just to have a place to put them, but capacity itself wasn't the issue. After a couple of weeks, they cleared out the adults as they worked through patient load and then moved the Peds unit back into their original spaces.