The “stale air” phenomenon afflicts oboists (sometimes clarinetists and others). It can be hard to relate to if you haven’t experienced it.
Here’s how it happens. (The “math” and “science” here are very simplified for clarity.)
The oboist breathes in a lungful of air. The air is about 20% oxygen and 80% other gases. The oboist’s body starts absorbing the oxygen and replacing it with carbon dioxide.
The oboist starts to play. The oboe reed has a small opening in it, so the air leaves the oboist’s lungs slowly.
A few moments later, the oboist’s body has replaced the oxygen with carbon dioxide. But the player’s lungs are still, say, 50% full. The oboist’s brain needs oxygen and starts urgently demanding a breath.
The oboist tries, but his or her lungs are still 50% full of “stale” (un-oxygenated) air. He or she can only get a half-breath of “fresh” oxygen-rich air. Now the player’s lungs contain 10% oxygen, which isn’t going to last long.
This cycle repeats a few times while the oboist gets more and more uncomfortable.
The oboist finally panics and quits playing to “reset” his or her breathing and get some oxygen.
A well-meaning educator sees the oboist struggling with breathing. He or she unhelpfully pencils in a few more breath marks. This is going to make the problem worse as the oboist takes even more unneeded breaths.
The solution to this is to figure out an outlet for the stale air. (Taking smaller breaths isn’t a great solution because it encourages weaker breath support.) In some cases it may be necessary to use a “breath” to actually exhale stale air. Then, after playing a little more, get a satisfying breath into emptier lungs. In other cases, it might be a better solution to do a quick out-in breath.
Stale air isn’t something that people encounter day-to-day. So it’s not well understood, sometimes even by oboists and other wind players who deal with it. Being aware of the problem makes it relatively simple to solve.