First I'll do a couple examples from feminism, since it is often tarred as academic wankery, and I feel more knowledgeable about it:

• Feminist theories say that movies underrepresent women, or represent them in relation to men. A simple count of the number of movies that pass the BechdelTest vs. it's male inverse shows this to be plainly true. In fact, the gap is breathtaking. Not only that, but this gap continues with movies released today, supporting the idea that only direct and conscious intervention can fix the gap and related iniquities in the portrayal of men and women in media.

• Feminist theory predicts that issues like female reproductive autonomy, education, and various categories of violence against women are strongly correlated. Statistics appear to show this is true (not indisputable; reporting and confounding factors exist).

As for postcolonialism, I'll give it a shot, though I'm not the best to speak on it:

• Postcolonial theory states that most of the institutions of formerly colonial nations (their media, the World Bank, etc.) fetishize the strong nationalist state and a capitalist economy with all the trappings (central banks, urbanization, progression from agrarian to industrial to service economy) that western nations have developed over the past two centuries, and will attempt to impose states where they can. Many argue that Western intervention in the Balkans and in Somalia bear this out.

• Postcolonial theory makes many other statements about development, like that postcolonial nations shouldn't try to emulate western paths of development (because they will result in poorer economic growth). Some of them are hotly disputed. However they are empirical.

• More broadly, postcolonialism says that for any intervention in a non-western nation, basing this intervention on methodology for western nation will yield worse results than building the approach up based on the ethnographic characteristics of that nation, despite the fact that international institutions seem to favor the former.