If I get a beer labelled 5% ABV, how accurate is that measurement likely to be?
tl;dr — By regulation, ±0.3-0.5%.
Chemically speaking, I'm not sure—maybe someone else can go into measuring techniques, alterations (continued fermentation?) during distribution, etc.
But countries and regions specify tolerances for error.
In the United States, according to Title 27: Alcohol, Tobacco, and Firearms, Part 7: Labeling and Advertising of Malt Beverages, §7.71:
In the EU,
There are two ways a brewery can measure the abv:
Both methods are exact and the accuracy is entirely due to the accuracy of the equipment used to measure.
The least accurate and precise is the case of (say) a microbrewer using a regular glass hydrometer to measure the OG and FG (method 1 above) - these read to 4 significant digits, with an accuracy of +/-1 SG at best. Without getting into the details, the accuracy then is about +/-5% of the final value for a regular strength beer. So for a 5% beer, that would be +/-0.25%.
With lab grade equipment, the accuracy of method 2 can be as good as 0.05%, since the specific gravities are measured with greater precision (and accuracy.) So our 5% beer would be with accuracy 0.0025%.