I've been following this and it sounded like they were going through a very long check list of possibilities. Trying one thing after another. And this whole thing about "we're getting close" was mostly that they were getting close to the end of the list of possibilities.
No, that's not it at all.
The properties of the Higgs are well known from theory, if it exists. Very roughly, the experiment was searching for evidence that a particle that exhibits those properties exists, with the major complication that its mass is unknown.
The "getting close" has nothing to do with any sort of process of elimination, but with observing enough events to have statistically sound evidence that such a particle was observed. This is tricky, because the Higgs cannot be detected. Instead, you have to work back from detections of decay products that may be several generations removed from the Higgs decay. Higgs creations are rare, even when you have the energy to make it possible, so it requires trillions of events to make the search sensitive to the presence or absence of the particle.
What they've found so far came from a search that was sensitive to a few of the particular ways a Higgs will decay, and they've found that a particle that decays in the same way that a Higgs would, with a mass of about 126 GeV, has been found with high significance. The reason for the hedging about whether it's the Higgs is that they've not yet tested all the properties known for the Higgs. So they've found a new particle, and it behaves as the Higgs would behave, but there are more properties to be tested.
Incidentally, at this point, the discovery turning out to be a standard model Higgs boson is the least interesting possibility available (barring this whole business turning out to be an outright error, I suppose). According to a report I read earlier today, there may have been some evidence that the particle was behaving slightly different from the simplest expectation, so I guess there's some hope that the particle physicists aren't obsolete quite yet.