People keep saying that around here. It's technically true (the best kind of true?) but it's not exactly how the real world works.
In the real world (meaning: actual companies, not startups with 2 weeks runway), to fire an individual (mass layoffs are different), you first put them On Notice (a Personal Improvement Plan saying "we, the company, believe you are not a good employee. you have X amount of time to improve"). Next, you wait 6 months. Then you review their progress against their "plan," you find any excuse for them not meeting their plan (plans can be subjective), then you say "Sorry, you're fired, here's a severance package."
So, firing an individual involves interdepartmental paperwork, getting legal potentially involved, and a (hopefully not trivial) payout.
Mass layoffs are essentially the same except you don't give 3-6 months notice and just immediately unload lots of people at a company, but hopefully still give them a payout.
In either case, I think why people throw around "at any time" is they want to convey you should have zero-loyalty towards any employer since they can unilaterally act against you. That is quite true. Never trust your employer to be benevolent towards you and always lookout for yourself before sacrificing things for the benefit of only the company.