I spotted this on the Rams website:
The same line is on the Wikipedia for NFL on TV, but the teams aren’t listed and the cite is broken. Anyone know what the 3 teams are?
Interesting, I wonder if these 3 special teams are the same every year or if they are rotating based on the quirks of the schedule. Wonder if they are hand picked at the start of the year based on profile/performance or if there’s a logistical issue that causes this.
It’s not the same teams year to year. It’s just the way the schedule works out due to the flexible scheduling at the end of the season - three teams will have played five times on prime time already and will have the potential to be flexed into that final game.
Something doesn’t jibe, though it might just be poor sentence structure.
The answer from the OP makes it seem like there are ALWAYS EXACTLY 3 teams with the potential for a 6th game. The ability to flex games makes for a logical explanation for why this is, but I don’t think that if that were the case that the numbers would be the same every year. In some years there’d be no teams with the ability to be flexed into a 6th game, and unless there’s another rule I’m unaware of it seems unlikely that there are always 3 teams who have played 5 primetime games prior to the flex period.
The Lions and the Cowboys traditionally play on Thanksgiving afternoon/evening. I wonder if part of that arrangement is that they still be eligible for five more prime time games.
This article (from April 2013) at NFL.com gives a little more detail.
[ul]
[li]5: Denver Broncos, New England Patriots, Atlanta Falcons, New York Giants, San Francisco 49ers and Washington Redskins[/li][li]4: Baltimore Ravens, Houston Texans, Indianapolis Colts, Pittsburgh Steelers, Chicago Bears, Dallas Cowboys, Green Bay Packers, New Orleans Saints and Seattle Seahawks[/li][li]**3: **Cincinnati Bengals, Miami Dolphins, San Diego Chargers and Minnesota Vikings [/li][li]2: New York Jets, Carolina Panthers, Philadelphia Eagles, St. Louis Rams and Tampa Bay Buccaneers[/li][li]1: Cleveland Browns, Buffalo Bills, Jacksonville Jaguars, Kansas City Chiefs, Oakland Raiders, Tennessee Titans, Arizona Cardinals and Detroit Lions [/li][/ul]It’s still not clear where the “maximum of 3 teams” that can be flex-scheduled to six comes from.
The article doesn’t say: is it implicit that a team cannot be flexed out of its only prime-time appearance? I haven’t checked the schedule - are all of those “one-timers’” games in the early part of the season, before flex scheduling kicks in?
Flex scheduling only applies to Sunday games. I think most of the one-timers are all Thursday games, so they can’t be flexed out of it.
The 2013 NFL Prime-time schedule can be viewed in its entirety here.
I’ve not checked to see that it gibes with the list I posted earlier.
That would do it. 
It appears that all of the one-timers except Oakland and Detroit have their games on the NFL Network (i.e., Thursday night). The Raiders and the Lions both appear on Monday night (weeks 3 and 15, respectively). As you point out, none of those games are subject to being flexed.
I’m not certain, but I’d bet there are three teams that already have 5 scheduled prime time games. All games week 17 may be moved to Sunday night no matter what, so perhaps those are the three teams that could play 6?
No, I believe there is currently no prime time game at all in week 17. The final Thursday and Monday night games are in week 16. The last game in week 17 ends by around 5:30~6pm PST that week. 13 games start at 1pm EST, 3 games 3 and a half hours later. I think they do this to equalize the interval to the first playoff round.