To answer this question, I gathered data from 2010-2015 and looked at how teams performed in the game following a non-standard week meaning they played on Thursday Night Football or Monday Night Football, or had a bye week.
Starting with Monday Night Football, if the short rest had an effect, we would expect to see a winning percentage significantly less than 50% for the game the following week. Over the 193 game results I gathered, the team playing on short rest went 91-102, a win percentage of 47.2%. A chi-squared analysis gives us a p-value of 0.6048. In other words, if the games results were determined by the flipping of a coin, we would see this much deviation from the expected win percentage about 60% of the time, meaning we cannot come to the conclusion that teams perform significantly worse on short rest. For the results to be significant, at the 5% level, we would have had to conduct the chi-square test using an expected win percentage of greater than 58% or less than 38%. Given the large sample size, it is reasonable to assume that teams coming off of Monday Night Football would’ve averaged out to about a 50% win percentage, but even if it was slightly different than 50%, the chi-square boundaries are far enough apart from 50% that the conclusions should still hold.
Moving on, I then examined how teams performed with extended rest in the week after playing on Thursday Night Football. Again, there was no real correlation as teams went 72-78, a p-value of 0.6242. Similar to Monday Night Football, the week-by-week was relatively inconclusive.
Finally, I looked at the true “mecca” of extended rests: the bye week. An entire week without a game, an extra seven days to get hurt players healthy and watch film to prepare for the next opponent, bye weeks are an oasis in the middle of the desert for struggling or beat up teams. Many teams complain about having their bye week early in the season when they feel like it doesn’t benefit them as much as say, week nine when they are halfway through the season and “could use the rest more.” Once again, when we look at the data, there is simply no correlation between bye week and performance the following week with an extra seven days of rest and preparation.
So what does this all mean? Basically, no team is significantly advantaged or disadvantaged by the amount of rest that they have coming into a game. Of course, specific scenarios like a star player not missing a game because of a bye week are exceptions to the rule, but otherwise, amount of rest is not a significant predictor of winning percentage.