Sort of related question, which may sound a bit rubbish since I don't really know the physics or their scientific method at that scale, but were they doing enough experiments that they'd have eventually said it couldn't exist if they never found it for some amount of time? Or was it always something that could only be proven, not disproven?
The Higgs was not guaranteed to be there, and its absence would have been obvious and discoverable. If they'd seen no signal by now everyone would likely be comfortable saying that there was no standard model Higgs.
However, SOMETHING needs to break electroweak symmetry---this is known because the W and Z gauge bosons are massive, and it isn't possible to formulate a consistent gauge theory with massive bosons without symmetry breaking. Whether it was the Higgs mechanism, or some technicolor (strongly-coupled, QCD-like) theory, or something else entirely was not well constrained before the machine turned on. With that in mind, it's clear that it might be that what they found is not just a standard model Higgs, but one of a number of higher-energy excitations. If that's true then the Standard Model is not, strictly speaking, correct, though it can be thought of as an effective field theory ( http://en.wikipedia.org/wiki/Effective_field_theory ).
> were they doing enough experiments that they'd have eventually said it couldn't exist if they never found it for some amount of time?
No, that's called "proving a negative" -- in most cases it's an impossible evidentiary burden. For example, no one will ever be able to say that Bigfoot doesn't exist for lack of evidence. This is why the null hypothesis is the default scientific precept -- that something without evidence is assumed not to exist (but by no means proven not to exist).
While it not possible to disprove the existence, of, say, unicorns. It is possible to say "I've searched the length and breadth of Central Park, looking for unicorns with 99.9% detection probability for each square meter of the park. I found no unicorns."
Whether you find this sufficient to reasonably exclude any unicorn hypotheses is up to you and will depend upon whatever unicorn hypothesis you wish to test. If the hypothesis is that the world should be uniformly populated with unicorns once every hundred meters, then the above observation should place strong constraints on the viability of the hypothesis, as about 340 unicorns should have been observed.
In the case of the Higgs, had the signal not popped up, they would have been able to exclude any Higgs-like particle over a huge range, which includes the now-claimed value. At the present mass, with the data up till now, they would have excluded a Standard Higgs to better than 5-sigma (less than a one in a million chance that they missed it).
Want to see the one of the peaks as it forms? (you have to trust that the scientists are doing the analysis right before they make this plot)
All scientists should take philosophy of science, IMHO, to avoid the misunderstanding of te scientific method that comes from dogmatic belief in it. I would guess most scientists don't know the difference between an inductive and deductive proof, both of which are crucial for scientific progress.
This graph shows how many pairs of photons were crated, classified by the energy.
There are a lot of ways to create a pair of photons, so there is a lot of background noise even if the Higgs Boson doesn't exist. This is roughly the dotted line (it's very difficult to see, because it's almost covered with the red line).
The black points are the actual measurements. They almost agree with the dotted line, except in the range of 125-130GeV where there is a bump. In that energy range there are more pairs of photons created than the expected quantity. The energy of the photons is essentially equal to the mass of the HB, but there are some dispersion because of the measurements errors and some quantum effects.
The red line is a simulation of how many pairs of photons would appear if the mass of the HB were 126.5GeV. They adjusted that parameter to get the best fit. This red line has a bump near 126.5GeV that is similar to the bump that the measurements have (black dots). The dotted line is very similar, so the red line cover it everywhere except in the bump range.
There is a lot of noise, so one possibility is that the bump is only a lucky streak, so they wait until they get a 5 sigma deviation, i.e. there is only 1/2000000 chance of getting a deviation as big as that form random fluctuations and noise. (The graph is old and shows only a 4.5 sigma deviation.)
The complete analysis is more complicated, but the general idea is that if there were no HB, they would get a different signal.
