That is an extremely complex question.
When you say the Japanese 'have' the tightest tolerances, you are asking a different question than whether the Japanese are capable of holding the tightest tolerances.
The answer to the first part is no--the Japanese developed very good standards, JIS that are comparable to ANSI (American National), DIN (Deutsche), etc.. The point of standards is to make parts that perform how they are supposed to, regardless of where in the world they are made.
The second part of the question is the really interesting one, and I don't have enough knowledge to really say one way or another. My personal experience is that most of the manufacturing countries are capable of about the same--Germany, Japan, and USA probably lead, but even countries like Taiwan, Brazil, South Korea, and India are producing outstanding parts.
What you heard is really a product of the 1960's through the 1990's (???
@Cornerman might know better and is welcomed to chime in). After WWII decimated Japanese production, they started fresh and adopted processes to consistently produce parts in spec in short periods of time--or at least well timed production. So, what really happened is that they lowered the cost of manufacturing by reducing the amount of parts that failed inspection. While doing this, they also refused to have huge stores of raw materials or finished parts, which led to lowered costs in shipping and storage.
It really is a difficult question that is a bit like fighting a Hydra--there are just too many variables to understand it all without seriously devoting a ton of time to researching it. That is really to get to the point of these paragraphs, which is to say that I've used precision parts from many countries, been part of projects that sent very high precision, very large machinery to many countries, but I only know enough to know that I don't know.