I don't know if it's anything but one great run. Others may make a different and more persuasive case, but the years when I think the team was absolutely unequivocally good enough to win the Super Bowl were: 2001, 2003, 2004, 2006*, 2007*, 2010, 2011*, 2014, 2015*, 2016. That's ten (10) and I think this is a conservative list as I've left off 2012 and 2013 and of course they were in the AFCCG each of those years. The years with the asterisk, which are four of the five seasons on that list they didn't win the Super Bowl, were years they were a single play from either winning the SB or going to a SB they likely would have won.
I think there's a reasonable argument to be made that in 2003, 2004, 2007, 2010, 2011, 2014, 2015, and 2016 the Pats were the best team in the NFL.