Short answer: this depends on how you define the White House's objectives.
To summarize the underlying story, there has been much back-and-forth in the on-line press today relating to a David Axelrod statement that the White House accepts the reality of the world we live in, a potentially anodyne principle that it more controversially interprets as meaning it can't get the expiring individual tax rate cuts below the top bracket extended permanently, while not also signing on to permanent extension of the top-rate tax cuts (which the White House, I gather and hope, remains unwilling to do).
The upshot appears to be that all of the tax cuts will be extended for just two years, presumably with the Republicans' agreement if the only alternative is to let them all expire.
This is indeed a cave-in on the White House goal of separating the top-rate tax cuts from all the rest. But it appeared clear that the Republicans were willing to go to the mat on this by playing out the chicken game in which the tax cuts expire and each side blames the other. Plus, it was far from clear how many Congressional Democrats would back the White House on this one. And if the impasse and tax rate increase hurt the economy, not only would this be bad for its own sake, but the White House might pay the main political price despite all efforts to blame the Republicans.
But suppose the White House didn't really want to extend any of the tax cuts permanently, but was merely playing defense (going all the way back to the 2008 election) in asserting the contrary. I'd certainly like to believe this was so, given the fiscal insanity at this point of losing trillions in long-term revenue. If so, then getting an excuse not to extend any of them permanently is actually a potential policy improvement over the stated White House position (with political cover being provided by the dispute over the top bracket). And indeed it's arguably better policy than letting the tax cuts expire right away, given the need for fiscal stimulus and the zero prospect of administering it by better-targeted means.
Of course, this raises the question of what will happen (and whose political interests will be served) when the tax cuts' expiration comes up all over again in 2013. The issue will surely feature prominently in the 2012 Presidential election, and Obama will likely have to repeat his 2008 pledge about the rate cuts below the top bracket (no matter how dire our long-term budgetary prospects). Plus, the concession reinforces an all-too-accurate (as well as self-reinforcing and politically costly) perception that the Democrats, as always, remain less willing than the Republicans to play their side of the ongoing chicken games with high determination and vigor.
My bottom line: subjectively speaking they caved, this was probably a realistic choice, but the long-term policy outcome could end up being better than if they had gotten the deal they wanted (2 year extension for the top bracket, permanent for everything else) - depending, however, on how this plays out in 2012 and 2013.
It would be nice just once in a while, however, to see the White House play its cards more aggressively. There will probably be chicken games galore in the first half of 2011, involving Republican threats to shut down the government 1995-style and/or to cause a temporary default by refusing to increase the debt limit, unless they get massive concessions that probably start but don't end with repealing or at least defunding healthcare. And if the White House enters these battles without any credibility regarding its willingness to stand firm, then the chance of a mutually costly miscalculation (or else of abject surrender on its part) becomes all the greater.