Try searching for the exact same flight, but with one person and see if the fare is low again. It could be that only one or two tickets were purchased at the cheaper fare, and your search for multiple tickets all together are moving it up into a higher fare.

However be careful booking tickets separately. In the event of a cancelation, weather issue, mechanical issue, or other IROPS, they might not keep the separate reservations together. And if one person is elite, then their benefits wouldn't transfer over to the separately booked group.

Here's an example:
11/28 SFO to BNA on the 8:45a flight, there are 5 seats available on a K fare, and 6 seats available on a Q fare. If you search for 5 people, it will give you $320.60 per person, all booked in K. If you search for 6 people, it will give you $349.60 per person, all on a Q fare. Even though there are 5 cheaper seats available, it doesn't put 5 in the cheaper seats and one in the more expensive seat.

This isn't necessarily the case, but it's a possibility and may give you the option to get some less expensive tickets.