The Pats have been a dominant team in the last 10 years, but they're also a team that hasn't won a Super Bowl in 8 years.
I'm a Yankees fan as well as a Giants fan. The Yankee won 4 rings from 96-2000. Then they absolutely dominated the next ten years in terms of regular season wins, playoff appearances, division titles etc....it's not even up for debate. BUT, they won only ONE more WS from 00-09, despite 3 WS appearances. As a Yankees fan, I don't brag about it. They still lost in the playoffs.
Since the Pats last Super Bowl win, the Giants and Steelers have won two. So, yeah, the Pats have been successful, but it's the hardware that counts, and it's been long time since they've acquired any.
Look at it this way, I'll bet any Pats fan would trade 4 of the last 8 seasons of making the playoffs for adding two more trophies.