The allowed duration of a stopover is defined by the type of ticket and (therefore) the airline. I've never tried to get such a long stopover, but I can imagine it would be possible.
However, this is as much an alternative to a RTW ticket as an actual RTW ticket is. On a typical flight between two locations, you won't have more than one stopover. Unless you've then got an open jaw ticket, you can only have a stopover in one location.
If you want to have stopovers in multiple locations, you indeed need to get multiple tickets, which would quite defeat the purpose of using stopovers to avoid getting a RTW ticket.
Furthermore, if you're switching airplanes on your stopover location, you have to pay airport taxes. As RTW tickets are mostly (though not completely) priced on the distance traveled, with airport taxes added on top, there is no inherent difference between an RTW ticket and a flight with stopovers.
Of course, the above is a general observation and doesn't say anything about any deals any airline might have for whatever reason.