It's odd to me that so many people argue about whether the Patriots are a dynasty.
Does it really matter? They have x wins in the last y games, a certain number of Super Bowl wins, and all the rest. There are numerous factors that can be used to measure the team. The word "dynasty" is merely a word used to simplify all those other factors into a single one. It's a convenience, like all symbols and labels.
So two points follow: the first is that whether they are a dynasty is irrelevant. If you win three Super Bowls in four years, and you're not a dynasty, does that diminish your accomplishments? Of course not.
The second is that because the point of the word is to simplify a complex set of data -- and, by spending a significant amount of time arguing about it, you're only adding to the complexity -- that if you have to spend much time arguing it, the purpose of the word is defeated and the answer to the question is effectively No.