The above is valid provided you are able to produce proof or documentation of onward travel and that the connection is within airport minimum connect times, is within the same day and meets local country restrictions.
That little paragraph is so important and, very often, the whole reason a passenger makes two separate bookings is because the connection fails the MCT criteria!
The MCT is extremely important as even 5 minutes means the difference between your bag travelling with you or missing your onward flight and perhaps getting lost altogether!