The 23:59 rule is very much a thing, and a great way to tack on a “free day” in whatever city you’re connecting. This can be especially useful if you’re flying internationally through a major gateway like, say, NYC or Chicago.
I’ve only ever done this with award travel, but my understanding is that whether it’s a paid fare or an award, if an airline’s booking system will recognize the lengthy layover and ticket it, then you’re good to go.
A couple of important caveats:
You’ll probably have to find a place to store your bags, though depending on your final destination and the airline(s) involved, you may be able to check them all the way through. A lot of bigger airports have luggage storage facilities.
If overnight lodging is involved in your layover, you’ll have to front the cost yourself. However, some international airlines will cover your hotel in certain cases, such as if the long layover can’t be helped. (Say, for example, your layover is in Istanbul and your flight gets there at 8 p.m., but the only flight within 24 hours to your final destination doesn’t leave until 6 p.m. the next day.)
After 24 hours, it becomes a stopover (rather than a layover) and generally costs more money, though certain airlines also offer free stopovers. Icelandair is one such example, as I recall.
Technically, you could string more than one long layover (23:59 or less) onto an itinerary — again, if the airline’s ticketing system allows it — but I tried that once and wouldn’t recommend it. At a certain point, you just want to be where you’re going!