I'm sure there are people who might do something like this on rare occasions. It seems naive to assume otherwise. But generally speaking, no. If you're using the "inferior" item, there's a reason.
I know a lot of people who have a nice, fast car they drive on dates, to social gatherings, on a road trip, etc., but drive their $500 clunker back and forth to work, while using their nice car as a "backup" when the clunker inevitably breaks down occasionally. But the fact remains that the clunker is better in one critical area: using it means you aren't devaluing your nice vehicle when you aren't getting anything out of it.
The same thing applies to everything else listed so far.
- A mistress may be superior to a wife in many respects, but clearly there's some reason the guy is still married. That one respect is the crucial area the mistress is not better. Maybe the wife cooks and the mistress doesn't. Maybe the wife makes six figures a year. Maybe it's just socially acceptable to have a wife and this guy cares about that kind of thing.
- A 4G modem is technically superior to a 3G modem so it's the default even if actual bandwidth doesn't need it. If reliability were really an issue, they'd switch to the 3G modems as primary. So really, 4G and 3G are comparable in practice, and there's no reason to prefer one over the other.
- Software developers are using the beta stuff because you can't develop for it unless you use it. Using the current, stable release of a driver for your upcoming software means by the time your software is finished, it's out of date. The beta driver is therefore better in the crucial aspect that it gives your product better longevity.
- Cloud storage, or other backup devices, are better for reliability, but are worse in crucial aspects like speed, offline availability, latency, etc. Imagine level load times if you were playing a game completely off the cloud.
- One car may be as fast as 5 horses, but it's not as fast as 50 horses. Also, it might be far more expensive in this village's economy, so they would save it for when it's needed, not just a little nicer.
In all cases, the backup has some critical reason why it's the backup and not the primary.
The only exceptions I can really think of are due to ignorance. A guy with R9 270 and R9 380 graphics cards might stick the 380 in his computer and put the 270 on the shelf as a backup because he assumes higher numbers mean more power, despite the 270 being substantially faster. When we were kids, my sister would pass in 5th gear instead of 4th, because she thought higher gears meant going faster without understanding the difference between top speed and acceleration. The 3G vs. 4G issue could be a result of this: some boss guy assumes 4G must be better and makes people use that instead of the older 3G that's better for their specific application.
Another exception that's not really an exception is laziness. Maybe the new system is faster and would technically save me lots of time in the long run, but I really don't want to bother learning how to use it. So yeah, it's superior in any externally-objective tests, but it's still locally inferior because that requires me to put forth a lot of effort right now, instead of putting in a little more effort each day that adds up to more total effort.