Can a theory be proven by an experiment (or series of) that sustains it? This is a question that I've always had but it wasn't until the recent news regarding the BICEP2 experiment that I really gave thought about it (please note that the question is general and not particular to the theories involved in that experiment).
I think a series of experiments or experimental data that conforms to a theory increases the odds that the theory is accurate, that through the theory things can be explained or the outcome of future things (e.g. experiment results) predicted given certain properties or conditions.
Kind of like in statistics and the concept of degree of confidence, that when more experiments sustain a theory the degree of confidence which holds the theory true is raised.
So... is it impossible to ultimately prove a theory? Is it eventually accepted as accurate or proven if the degree of confidence is high enough or is it actually possible to prove a theory by several different repeatable experiments that conform to it?
 A: In physics in contrast to mathematics,  it has no meaning to ask for proofs of the correctness of theories. Theories can only be falsified. Each new experiment testing a theory either validates it ( which means predictions are correct up to errors), or falsifies it.
In mathematics one starts with axioms, develops and proves theorems and can then know that the problems solved can end with a QED. The whole theory hangs together and cannot be falsified if correctly created. The axioms of the theory though can be falsified, as for example "two parallel lines never meet" in Euclidian Geometry, becomes invalid in spherical geometry and new axioms are postulated for a new coherent and self contained geometry.
Theories in physics rely on mathematical theories for their models, which can be used to predict outcomes for experiments only if specific to physics postulates are imposed on the mathematics. A "physics theory" thus consists of a self consistent mathematical model on which postulates are imposed in order to interpret the calculations relevant to a physical experiment/situation. 
example: Newton's laws. They are imposed over the mathematics using differential equations and integrations completely valid and unfalsifiable in the mathematical theory of real numbers. When experiments with very high energy particles gave inconsistent results, Newtons laws(theory)  were falsified for these high energies and the special relativity postulates were imposed on mathematical models, no longer  with real numbers only, but with complex ones too.
The postulates in physical theories are as extra axioms, but  the mathematical model's  validity is independent of the postulates.The postulates  guide the interpretation of the calculations, so as  to pick up solutions relevant to the physical boundary conditions, and thus  predict experimental outcomes. A failed prediction invalidates  the postulates, i.e. the "physical theory", which will have to be modified or rejected. 
Thus even one accurate measurement can falsify a "physical theory" but many measurements with statistical validity are necessary to  validate it.
In physics there is continuity, so each mathematical model has a region of physical validity and as the frameworks ( dimensions in space and momentum) change new postulates and models are proposed to become validated in the new region.
A: No, you never prove theories [ie, models] in science. The goal is never to be right. The goal is to have a model that makes definite predictions about many observations, each of which can be confirmed. Models that have achieved this success don't get overthrown just because someone points out that the original model hasn't been proven, they only get thrown out when someone either comes up with a better model or finds a specific prediction of the model that disagrees with what is observed.
I really like this quote by von Neumann:

The sciences do not try to explain, they hardly even try to interpret,
  they mainly make models.

"Method in the Physical Sciences", in The Unity of Knowledge (1955), ed. L. G. Leary (Doubleday & Co., New York), p. 157
A: This is really just a footnote to Anna's answer, but it got a bit long for a comment.
Physicists are in the unenviable position of knowing that every theory we propose is wrong as soon as we propose it. With the possible exception of canonical quantisation I can't think of a single physical theory that isn't known or at least strongly suspected to be wrong.
This is in the nature of the beast. To develop a theory physicists look around so see what's happening, make a few assumptions to simplify the problem then propose a mathematical model to describe it. The problem is that the assumptions that go into the theory always mean there are some circumstances under which it fails. The example of Newton's laws that Anna gives is an excellent example of this. It's a mathematically elegant theory that fails at high velocities. So we use Special Relativity instead, which includes Newton's laws as a subset, but that fails to describe gravity. So we use General Relativity instead, which includes Special Relativity as a subset, but that isn't compatible with quantum mechanics. So we use String Theory instead, which includes General Relativity as a subset, and so on, and on, and on.
But all is not lost, because repeated experiments establish the set of conditions under which a theory is reliable, and clarifies the boundaries beyond which it fails. The excitement about the BICEP2 data is that it provides experimental tests in areas we weren't able to test before.
