The question is partly rhetorical. You’re supposed to get affronted and say, “No, of course you can’t. Everyone should know that. You can only prove a theory false.”
But to dig into the question properly, you have to define “theory” and consider the difference between closed and open systems. The word “theory” is used in many different senses; as Marcelo Gleiser points out, you must observe its context to understand its meaning.
A theory is more than a hypothesis, which in turn is more than a guess or one-time prediction. A hypothesis is a tentative explanation of a general phenomenon: for instance, “Rain occurs more frequently where land is cultivated.” This can later be developed into a theory (e.g., “Rain follows the plow“). Sometimes there is disagreement over whether something should be called a hypothesis or a theory–but the distinction remains.
A theory must be well substantiated, have broad application, and explain a general phenomenon or class of phenomena. When a theory takes the form of an apparently unwavering principle, it may be called a law.
If I say, “There is oil on Mars,” I am not positing a theory; I am just making an untested assertion. If I were to say instead, “Where rocks on a planet show chemical composition X, there is oil beneath the planet’s crust,” that would be a hypothesis; with strong basis and explanation, it could be a theory. If I were to find a unifying principle predicting the presence or absence of oil, I might call it a law. Any of these can be refuted: assertion, hypothesis, theory, and law. Only the assertion can be proven true, in the case that oil is found on Mars. Even then, there are caveats.
Now, growing up in a family of mathematicians, I assumed in childhood that you could prove a “theory” true (provided that your axioms were true). Because mathematics works within a closed system, you can work logically from axioms to conclusions and thereby demonstrate that the latter proceed from the former. (In mathematics, it is a “theorem” that you prove, not a “theory.” The word “theory” in mathematics usually refers to a body of knowledge. The usage is not entirely consistent, though; one hears of Ramanujan’s theory of primes, for instance.)
In the natural sciences, you never have a completely closed system, except in the theoretical fields. This is where things get tricky. In theoretical physics, for instance, you can determine from your axioms and laws how your model will behave. Models do not completely match the natural world, though. In the natural world, there are mitigating factors (and the truth of the axioms matters a great deal).
The social sciences may be the farthest from any closed system–because so many factors, past and present, can influence human behavior. You usually work with high degrees of uncertainty. What do you do? Do you just give up? There are those who believe the social sciences are pure nonsense, but I am not among them. I favor efforts to make sense of our lives from the standpoints of many different fields–including philosophy, literature, mathematics, theology, languages, statistics, physics, psychology, and more. Each field contributes in some way to our understanding.
But how does one work with so much uncertainty? First of all, enjoy it now and then. It would be a dreary world if we could figure everything out. One doesn’t have to be perpetually cheery about it, but one can take courage from it. Second, find ways of working with degrees of uncertainty–not treating all uncertainty as alike, but determining which are greater than others. Models in the social sciences can bring much insight; one must just take care to observe their divergence from actual phenomena.
It’s important, when doing this, to avoid the null hypothesis fallacy. Some might say, “Well, I can’t prove a theory true, but I can prove its negation false, and that’s essentially evidence for the theory.” No, it isn’t. The two are not the same. When proving the negation false, one does not win evidence for the theory; one is still firmly fixed in the wobbles of doubt. One must figure out how to view the doubt clearly.
In any case, the answer to the initial question is double. If you are working within a closed system, you can prove a theory (or theorem) true; within an open system, you cannot. However, in the latter case there is still much you can learn.
Hypothesis: A proposed explanation for the way something works. (It is more than a “guess”; it must have a basis in evidence and reasoning, and it must be testable.)
Theory: A hypothesis that has been tested, substantiated, and extended, and that applies broadly to a natural phenomenon or class of phenomena.
Law: Like a theory, but unified into a general principle that unerringly explains a phenomenon (to the best of our knowledge and understanding).
Model: A representation of a real-world phenomenon, designed to assist with observation, testing, and explanation.
Note: I revised this piece substantially after posting it. In particular, I clarified the terms, changed the examples, and added some links. I cut the part about literature, since it needs a post of its own.