I recommend Russ’s recent EconTalk with Morten Jerven.  Jerven’s forthcoming book, Poor Numbers, shows that GDP estimates in sub-Saharan Africa are fraught with error. Jerven, who visited many statistics offices in the region firsthand, concludes that poorly staffed offices and ever-shifting measurement standards give us reason to doubt the quality of these data.  The whole hour is filled with insights, do listen.  

Jerven makes it clear that low quality data aren’t just a problem for sub-Saharan Africa–the problem is endemic across less developed countries.  Low productivity in the private sector typically predicts low quality in the public sector–and that predicts low quality government statistics.  

Independently, Russ offers another potential problem: The temptation of governments to report bad news to make sure the foreign aid keeps pouring in.  It would be valuable to test this empirically but until somebody comes up with a solid test, let’s just stick to theory: governments must face some temptation to report news that differs from reality.  Some governments will be tempted to report news that’s worse than the truth, some that’s better than the truth.  
That temptation to tell excessively good news surely exists in the rich, democratic countries, at least most of the time.  But the civil services in rich countries are immune to crude manipulations of GDP statistics, aren’t they?  Apparently not. Consider this study of GDP data in the G-7 club of rich economies:

…in the Japanese data, the last GDP growth rate announced before national elections is systematically about one percentage point (at an annual rate) more optimistic than in other quarters. This result is robust for Japan, but the election effect is not found in any other country. 
Is there a way out?  Is there some independent way for mere observers like us to find out what’s really happening? Yes, there is. TC:
Travel is the starting point of learning social science. 
And I’d add “…and of learning GDP per capita.”  Not the only way–the financial sector has a strong incentive to find out which regions of the world have dollar bills on the sidewalk, so keep an eye on their reports–but checking out a few different countries or asking questions of people who have visited these places is worth quite a lot.  
Journalism, anthropology, talking to friends, seeing places yourself (especially outside tourist-friendly cities): There are quite a few ways around the bad data problem.  True for rich countries as well as poor. 
Does the place seem as rich as the statistics say?  If your eyes disagree with the official data, maybe it’s the data that are wrong.