My best answer is that it's insanely improbable. The good news is it looks like two of the three players were only out for the game, but still I don't remember having seen one team have three injuries in a game off the top of my head, and I carried a level 2 doctor for many seasons.
Taping injuries could have helped - my best guess is the two guys who were injured but are healthy today might have come back to start the next quarter (if it didn't happen in the 4th), but it could be that this would have changed the whole pattern of randomness and instead of having three injuries, you'd have had a guy with 11 FT go 0-8 from the line.