I saw it first hand. My manager was pushing everyone to get promoted to level 5, of the few who didn't, one was put on a pip then fired, the others worked like madmen after seeing that, still didn't get promoted and somehow moved into another group. They still haven't gotten promoted but they work insanely hard/ aren't the same people. I guess it depends on the group, but I was told that timelines for promotion to level 5 are an unwritten rule.

there's no 'unwritten rule' for promotion timelines

It was written for a while - the promotion guidelines used to state that the expectation was that all engineers were expected to get to level 5 within "a reasonable amount of time" (generally 2-3 years between promotions), and that managers should identify roadblocks to continued growth for people stuck at level 3/4. I don't think that the "or out" part was ever specified, but that's the implication. Level 5 was continued a terminal level for folks who didn't want to move into management, roughly equivalent to tenure; at that point, you've paid off your value to the company, and can be trusted to pick your own projects.

I heard less and less about this policy over my time in the company - it was stated explicitly my first few perf cycles in 2009/2010, but I don't think I could recall it ever being mentioned after around 2012. And it was always very sporadically enforced; the longest-serving level 4 I knew had been with the company for 11 years.

It's actually a written rule. Look up the job descriptions for SWE2's and 3's and compare them to those for higher levels. It specifically says that you are expected to advance.