No one beats the Solow Model

The Solow model says many things, but one of the highly observable ones is that the income levels of poor countries will tend to converge towards that of rich countries.  More to the point, it'll converge towards those rich countries with similar characteristics.
The point?
Ah. The point being that if the country that is going to eat your shirt (China! Japan before that! Germany! etc.) is starting from a lower baseline than you, odds are that the closer it gets to you, the more it slows down.
Or, to put it differently, you can't assume that a country is going to grow at 8% for the next 20 years just because a country grew at 8% for the last 20 years.  So yeah, 20 years from now, China isn't going to have flying cars and free ponies.
Noah Smith has a pretty good take on this based on the question - What happened to Japan in 1991?
As he puts it

Basically, by 1990, Japan had caught up to the richest large nations in terms of per capita GDP. The only way for Japan to have continued at its previous high rate of growth post-1990 would be for either A) an unprecedented technological boom to power a rapid expansion among all the world's rich countries, or B) for Japan's productivity to significantly exceed that of the other rich countries. In other words, anyone who forecasted continued rapid Japanese growth in 1990 was predicting that Japan was capable of doing far better than the other countries of the world, and indeed that this was the most likely outcome.

Now, let's look at Japan right now. Wikipedia tells us that Japan's per capita GDP, in nominal terms, is $45,900. That compares with $44,500 for Germany, $43,100 for France, and $39,600 for the UK. I picked Germany, France, and the UK because these are other rich developed nations with populations between 50 and 150 million and growth rates similar to Japan's. In other words, in nominal terms, Japan is richer, per person, than any comparable country. I suspect that what difference exists is due to labor inputs, since Japan does not force its citizens to take lots of time off of work the way Germany and France do.

In short, its bang dead to rights as far as the Solow model is concerned, no surprises there.

What could forecasters in 1990 have been smoking that made them see this as anything other than the inevitable outcome? I suggest two things: 1) dumb trend projection, and 2) attribution error.

Dumb trend projection is people's tendency to view trends as structural. Examples of this include: "Housing prices have never fallen; hence they will never fall." "This stock returned 17.2% over the past three decades; hence it will continue to do so." "The center of gravity of the global economy is inevitably shifting to Asia." And so forth. But in reality, past performance is no guarantee of future results. Or, to put it more pithily, "The trend is your friend till the bend at the end."

Attribution error is our tendency to attribute phenomena to the wrong causes in certain reliable ways. One of these is that we tend to attribute outcomes to fixed person-specific characteristics rather than to circumstance and situation. In Japan's case, this means that people thought that Japan was growing fast in the 70s and 80s because of Japanese culture, superior Japanese government, or - and I suspect that this was more significant than people will admit - the inborn superpowers of the Japanese race.

To summarize, don't expect South Korea and China to keep growing at the current break-neck pace, no matter how much we try to convince ourselves that this time its different.