World Snooker never disclose the exact size (no idea why), although snooker fans seem agreed it is roughly 84 mm (which is something like 3.25 inches I think).
This however is fairly representative for a club table:
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Actually I was wrong about this being typical of club pockets, this article is quite interesting, it seems club pockets are closer to 3.5 inches typically (frankly I'm sure I'd still be rubbish if they were 4 inches ;-)!
http://gclbilliards.com/tag/snooker/