Based on this year's schedule it seems that an NFL team's bye can come anywhere from the 4th week of the season to the 12th. Is there either conventional wisdom or statistical study of which weeks are most advantageous? I would think that a week 4 bye would be a bad deal: few players are going to be so banged up at that point that the week of makes a big different for them.
There are many different opinions about this, and there is no consensus. That being said, circumstances are more important than the actual number. In general, there is no particular week number that is any better than any other for all teams. However, bye weeks can range anywhere from being extremely advantageous to somewhat of a set back depending on the particular circumstances of a particular team at that particular time. As you mentioned already, injuries are one factor. However, it does not follow that the later byes are always best for injuries. Many early season injuries are healed up by the time a week 12 bye would roll around. In addition, suspensions and healing from off-season surgeries are more common in the beginning of the season. One thing that teams look at regarding whether a bye week is timely or not is whether they are hot at the time or not. If the team is in the middle of a winning streak, they might feel that the bye week causes them to lose momentum. Similarly, if the team has just come off a really huge disappointing loss, this could also be considered non-advantageous since team morale is low during the practices. Other teams/coaches may disagree, however, reasoning that after a huge loss is a good time to make adjustments. Going back to whether a team is hot or not going into a bye week, teams prefer to be hot towards the end of the season rather than the beginning of the season since being hot at the end more frequently leads to playoff births/victories and championships. That being said, a later bye week could be detrimental to a legitimate playoff contender.
However, teams do not get to pick when they are hot or not so this does not rule out the late bye week as being the most advantageous for a playoff caliber team. In my opinion, the best time to have a bye week is after a respectable loss or a win that is not part of any considerable streak or when a team is just struggling on offense or defense and could use some extra coaching, practice, and re-evaluation of strategy. Whether or not a team is injury laden when going into a bye week is indeed the number one factor of whether or not their bye will be productive, but the other factors mentioned are also important.
One more thing to consider might be win-loss percentages. If your team has a winning record early, the bye week may result in the team being higher in the standings if their divisional rivals have tough schedules and later bye weeks. In the end, this will make no difference regarding a playoff birth, but teams at the top of their divisions typically get much more positive attention from the media which often results in better morale for the team and greater sales of team merchandise. This is really a minor issue though.
In summary, there are a plethora of factors that affect the productivity of a bye week, and when the bye is most beneficial depends on the particular circumstances that the team is in during the season. Unfortunately, there is no way to plan around this, so each team just has to take their bye whenever it is scheduled. If the bye works out really well, nobody says anything about it. If the bye was not very productive, then the fans and teams often complain that they got screwed by the timing. The truth is, it is all just a toss of the dice anyway.
Relying on pro-football-reference.com, I went year-by-year to get some data.
Season Winner Bye Loser Bye
2013-14 SEA 12 DEN 9
2012-13 BAL 8 SF 9
2011-12 NYG 7 NE 7
2010-11 GB 10 PIT 5
2009-10 NO 5 IND 6
2008-09 PIT 6 AZ 7
2007-08 NYG 9 NE 10
2006-07 IND 6 CHI 7 <--Final season with possible week 3 bye
2005-06 PIT 4 SEA 8
2004-05 NE 3 PHI 5
2003-04 NE 10 CAR 3
2002-03 TB 10 OAK 3
2001-02 NE 16 STL 8
2000-01 BAL 14 NYG 8
1999-00 STL 2 TEN 7
1998-99 DEN 7 ATL 3
1997-98 DEN 7 GB 8
1996-97 GB 8 NE 5
1995-96 DAL 8 PIT 7
1994-95 SF 9 SD 5
Before we start, there are a couple things to consider:
- Bye weeks have only been restricted to weeks 4-12 since the 2007-2008 season. Before that, bye weeks seem to have been as early as week 3 or (until 2001-2002) as late as week 16.
- Rules of scheduling changed after the 2001-2002 season, when the NFL began their current schema of 4 divisions of 4 teams.
- 1993-94 was the final season when teams had 2 bye weeks. I've cut my research off here because this muddies things up significantly.
In the years when the bye week had to be between weeks 4 and 12, the Super Bowl winning team's average bye week is just over 8, while the losing team's average is about 7.5. Oddly enough, the winning teams have had a bye in each of the possible weeks except for weeks 4 and 11.
Walking further back through time, we see a 3-year stretch when a team with an abnormally-early week 3 bye made it to the Super Bowl: 2002-03 Oakland Raiders, 2003-04 Carolina Panthers, and 2004-05 New England Patriots. Of these three, only the Patriots went on to win that year's Super Bowl.
Eyeballing the data, a bye in weeks 7, 8, or 9 seems like a benefit as the majority of Super Bowls have had at least one team from one of those three weeks; this breaks the season up into two approximately balanced halves, similar to the All-Star Break in Major League Baseball.
I think this depends on how the teams handle their practices. If you are playing for a Tom Coughlin, Chip Kelly, the Harbaughs, or any other coach running a a tough training camp and practices you want your bye week on the early side. Given that the NFL severely limited hitting in practices during the year, most of the major hitting is done in the first 6 weeks (counting camp). That is where you will see most injuries and player fatigue.
So a team playing for a tough coach will love a bye in weeks 4-7.
While on the other hand there are coaches who run softer practices and rely on athletes being able to perform without going through daily hitting. They probably wouldn't care as much about the bye weeks and would probably would prefer something that broke up the season well, maybe in the 7-9 range.
Also a younger team would probably want an earlier bye week (STL, SEA, TB). It is hard for coaches to change schemes during a normal week. But during a bye week they could install new packages and rework things. For a more veteran team (DEN, NE, GB) they probably don't care as much about this and probably have the ability to install smaller packages during a normal week.
There are certainly better and worse weeks, but it's much more dependent on a team's schedule and injuries than the week number. It's much more valuable to have a bye in the midst of the roughest matchups than a bye before facing the worst team in the league.