As we all know, it depends entirely on the agent you get. The rules are quite specific now about where the stopover must occur:

- TO DETERMINE ENROUTE STOPOVER DISPLAY AVAILABILITY FROM
ORIGIN TO FINAL DESTINATION. THE CONNECTION POINT
WHICH APPEARS IS WHERE THE ENROUTE STOPOVER SHOULD OCCUR

But there are agents who will understand that to go from San Francisco to Paris with a stopover in Denver should be OK on United since Denver is an obvious connecting point for United. Although not necessarily for Paris. If you ask for a stop in (for example) Oklahoma City, though, they would probably all say no.

As usual, the answer is to hang up and redial if you think your request is reasonable and you're looking for a different answer.