(NB. Playing on PS4 and I wouldn't say I have an extreme case of this problem, though it's bad enough that it's put my playthrough on hold. So please note that what follows does not apply to people who have more severe problems like going hours in total silence or who experience missing dialogue lines during banter).
I did some more experimenting today and I'm fairly certain now that there are a number of things that people aren't counting as banter but which the game does count as banter. For instance, location-specific one-liners (eg. comments on weather) seem to be counted by the game as party banter and they reset the semi-randomised banter timer. I also have a suspicion that quest banters reset the timer, even if they are just one-liners like "There might be something interesting here" or "This looks like a good place to set up camp". I don't know if combat dialogue interferes with it, but it's a possibility.
If I count one-liners as party banter then I typically have a companion say something about every 20-30 minutes, which seems to be at the upper end of the normal range. Unfortunately, if I manage to fire off a one-liner every 20-30 minutes several times in a row (which is not difficult to do) then I can easily go 2 or more hours without hearing any of the companions actually speak to each other. This happens more often than not during a typical play session and it means there is an overall paucity of two-way banter during the length of the game. (As an illustration, the companions can be paired in 36 different ways, eg. Solas and Cole, Solas and Dorian, Solas and Sera etc. With two hours between each banter that's 72 hours of solid adventuring just to hear each pair of companions banter with each other once.)
This is why I initially thought banter was working correctly for me; companions were speaking - but not to each other. Unfortunately, unlike two-way party banter, these one-liners do repeat to the point that it encourages you to eventually stop thinking of them as banter. What that's meant is that despite thinking banter was normal at first, I now find that I'm still getting Cassandra talking to Solas about how surprised she is that he's decided to join the Inquisition after I've already completed half the main quest, dozens of hours into the game. I could have fairly easily completed the game in that amount of time. (Worse yet, I only managed to fire off that banter after two hours of standing around waiting for it in Redcliffe, a spot I deliberately chose because I know that in Redcliffe neither companion seems to have any location-specific dialogue that might interfere with the firing of two-way banter...)
Oh well, at least it's progress of a sort.