de Mesquita is no more able to predict the future reliably than anyone else
There are two classes of forecasters: those who don't know and those who don't know they don't know. J K Galbaith
What de Mesquita claims is, strictly, not to be able to predict the future, but to predict the outcome of certain classes of events where the outcome is determined by some form of negotiation potentially amenable to Game Theoretic analysis and rational choice theory.
One of his most famous predictions was to correctly name the successor to Ayatollah Khomeni as Iran's Supreme Cleric (the story is told here):
His first foray into forecasting controversy took place in 1984, when he published an article in PS, the flagship journal of the American Political Science Association, predicting who would succeed Iran’s ruling Ayatollah Khomeini upon his death. He had developed a rudimentary forecasting model that was different from anything anyone had seen before in that it was not designed around one particular foreign-policy problem, but could be applied to any international conflict. "It was the first attempt at a general mathematical model of international conflict," he says. His model predicted that upon Khomeini’s death, an ayatollah named Hojatolislam Khamenei and an obscure junior cleric named Akbar Hashemi Rafsanjani would emerge to lead the country together. At the time, Rafsanjani was so little known that his name had yet to appear in the New York Times.
Even more improbably, Khomeini had already designated his successor, and it was neither Ayatollah Khamenei nor Rafsanjani. Khomeini's stature among Iran’s ruling clerics made it inconceivable that they would defy their leader’s choice. At the APSA meeting subsequent to the article’s publication, Bueno de Mesquita was roundly denounced as a quack by the Iran experts --- a charlatan peddling voodoo mathematics. "They said I was an idiot, basically. They said my work was evil, offensive, that it should be suppressed," he recalls. "It was a very difficult time in my career." Five years later, when Khomeini died, lo and behold, Iran’s fractious ruling clerics chose Ayatollah Khamenei and Hashemi Rafsanjani to jointly lead the country. At the next APSA meeting, the man who had been Bueno de Mesquita’s most vocal detractor raised his hand and publicly apologized to him.
The trouble is even this supposed great success is not a great as he or his acolytes make it seem. The surprising nature of the prediction looks a lot less clear when you read this (from a NY Times article published two year before de Mesquita's predicition (my emphasis):
None of this is likely to change much while Ayatollah Khomeini lives, in the view of diplomats in Teheran. But he is 83 years old, his health is frail, and the search for a successor has begun. A committee to decide will be elected on Dec. 10, but most politicians here seem to assume that a triumverate will emerge to run the country after he dies. It would include Hojatolislam Rafsanjani, who is regarded as the most able politician; President Khamenei, and Ayatollah Hussein Ali Montazeri, the Imam's designated heir, who is considered intellectually weak and insufficiently sophisticated by many politicians.
More recently de Mesquita made the prediction that the Arab Spring would not spread to Lybia (see this video starting about 1:50 into the segment):
...So it is unlikely to spread to those parts of the Middle East that have substantial oil wealth such as Saudia Arabia Libya and so forth...
This one didn't turn out so well.
We could test his predictive accuracy by forcing him to archive all his predictions about the future from today forward and checking them in a year or two. But right now we mostly have his word for the predictive accuracy as many of his predictive successes as told in his book were not made public before the event.
There are also theoretical reasons for doubting high predictive value. Anyone who has read, for example Taleb's The Black Swan, should appreciate that many major events are often influenced by incidents that are rare enough not to have happened before and therefore can't be included in existing probability estimates based on historic observations. No model based on rational choice or game theory can include such considerations.
I could go on and I could list more debunked predictions. But the job has already been done by others. For more detail and more examples I suggest looking here (I found many useful sources there).
In short: de Mesquita's predictions are not as reliable as his supporters think and even his successes are not a surprising as he claims when examined closely.