Don't think so. With the advent of what, Direct TV? You can see any team every game, so people who grew up, say in Seattle, but now live in Miami, can still be Seahawks fans. Before, Dallas was on the tube so much and so often, in addition to winning so much, that those Seattle people could become Cowboys fans just because they saw them so much.
And I'm not talking about bandwagon fans, of which there are many, who change allegiances at the drop of a hat, again because it's so easy to see the "hot" team now.
Of course, and speaking hopefully not just as a Cowboys fan, in my view there can never be a different America's Team. That is a title that was bestowed on Dallas by NFL Films (with the Cowboys agreeing to it of course). If you now say a team like the Patriots are now America's Team, you have to start qualifying it with what time frame you're speaking of.
And I really don't think any other team wants that title, it's well known that Dallas got, and continues to get, so much hate because of it that other teams won't want to deal with that.
For good or bad, the Dallas Cowboys will always be "America's Team", regardless of how well, or bad, they play....