Are the Bears suffering worse problems with injuries than other teams, or does it only seem that way to Chicago fans? All teams face injury, and good teams plan for injury. They plan for players being hurt, and they have a backup plan in place. Witness the Dallas Cowboys leading the NFC East despite the loss of their starting quarterback. However, it is fair to say that some teams have been hit worse by the injury bug than others.
First, there's injured reserve. At the time of this writing, 1156 "player games" have been lost to injured reserve in the 2016 season (in other words, more than a thousand times this season, individual players were unavailable for specific games because they were on IR). That's an average of 36 player games lost per team. The leaders in this category are the San Diego Charges, having lost 90 player games; they are followed by Baltimore Ravens at 72. In the NFC, and third overall in the NFL, the Chicago Bears have lost the most player games to injured reserve at 64, though the Saints with 61 are not far behind.
Second, there's the players who are simply "out," or "doubtful," or "questionable" without playing. The players lost to these lists are typically lost for less time, and the lists themselves are shorter, so only 469 player games have been missing from the NFL due to these injuries (about 15 per team). In this category, the Bears lead their division at 17, but come in fourth in the conference (the Cowboys lead at 22) and 10th in the NFL (the Cowboys are tied with the Steelers and Browns for the lead in the NFL).
Added together, the Bears have lost 81 player games only 5 games into the season, with (again) only the Chargers and the Ravens missing more players to injury than Chicago. That means that roughly 16 players per game have been unavailable because of injury.
Admittedly, simply counting the players lost to injury is an incomplete measure of the impact of those injuries. Still, the Bears contend here as well. Kyle Fuller, Hroniss Grasu, and Pernell McPhee all represent starters who have been lost to injury since before the season began. Regardless of how individual fans might feel about Jay Cutler, losing a starting quarterback to injury, for even a little bit, can have a dramatic impact on a team. However you want to look at it, the 2016 Bears season has had a disproportionate share of injuries so far.
Before fans raise their fists in anger that the Bears always seem to be injured, however, it's worth pointing out that last year was not nearly so bad in terms of player games lost. In fact, the Bears only lost 11 players per game to injury (that includes all three flavors of ‘might play' ranging from Probable to Doubtful, and it also includes ‘out', so long as the player in question did not play; it also includes the PUP list). The average team in the league last year was down 10 players per game.
In fact, the Bears only ranked 10th in total games lost last year, and three more teams were within a handful of games. The Patriots (281 total player games lost) and the Ravens (236) led the way for the league last year, with Washington and the Giants leading the NFC at 228 player games lost each (more than 14 players per game, well above the rate suffered by Chicago). Of all teams last year, only one lost more players per game to injury than the current rate suffered by the Bears.
As a side note, it is only fair to point out that Football Outsiders has a much more complex metric in which they adjust for the value and impact of players lost to injury. By that measure, the 2015 Bears were 5th in the league in terms of "Adjusted Games Lost" to injury. This could explain some of the anxiety felt by Bears fans in that while in terms of total number of players injured the Bears didn't suffer too badly last year, but that the players they lost were potentially impact players.
What do I conclude from all of this? The Bears, as a rule, might not be substantially more injury prone than other teams. On the other hand, the 2016 Bears really are suffering. Compared to the entire league, they have a lot of players too hurt to play, and a lot of those injured players are significant for a team that is still trying to rebuild its depth. If there is any positive to take away, it's that at least the backups are getting reps and experience.
*Unless otherwise noted, all statistics come from Pro Football Reference