Witness these two polls.
A Washington Post-ABC News poll in July 2012 -- more than three months before that year's presidential election -- showed just 6 percent of Americans said there was a "good chance" they would change their mind about their candidate of choice. That number was 10 percent in 2008 and 12 percent in 2004.
What's more, 19 percent said there was at least some chance they would change their mind (mandatory Dumb and Dumber link). That number was 25 percent in 2008 and 21 percent in 2004.
A Pew poll in 2012 showed a similar decline in genuinely persuadable voters. By April of that year, just 23 percent of people said they weren't certain voters for either Democrats or Republicans. That was down from 33 percent in 2008 but slightly higher than the 21 percent who said so in 2004. In the prior three presidential elections, that number was between 27 percent and 32 percent.
If 43 percent of Americans are truly independent, why are only half of them actually persuadable seven months before a presidential election? Sure, you can be a reliable Republican voter while not wanting to be a member of the party; but the point is that, for all intents and (electoral) purposes, and practically speaking, you are a Republican.
And then there's this chart, from Pew's political polarization study, which shows the shrinking ideological middle.
(Of course, just because people aren't as moderate as they used to be doesn't necessarily mean they are more partisan, but it stands to reason.)
You can look at any one of these charts and make the case they don't necessarily mean independents are on the decline -- just that swing voters and moderates are.
What you cannot say, though, is that genuinely independent American voters are on the upswing. About the only thing that's increasing is the self-delusion of independence.