"The Chinese use two brush strokes to write the word 'crisis.' One brush stroke stands for danger; the other for opportunity. In a crisis, be aware of the danger - but recognize the opportunity."
- John F. Kennedy (1917-1963), speech in Indianapolis, April 12, 1959
So, apparently, this is some popular misconception. This site goes into quite a bit of detail as to why this interpretation is completely wrong.
Like most Mandarin words, that for "crisis" (weiji) consists of two syllables that are written with two separate characters, wei and ji.
While it is true that weiji does indeed mean "crisis" and that the wei syllable of weiji does convey the notion of "danger," the ji syllable of weiji most definitely does not signify "opportunity."
The ji of weiji , in fact, means something like "incipient moment; crucial point (when something begins or changes)." Thus, a weiji is indeed a genuine crisis, a dangerous moment, a time when things start to go awry. A weiji indicates a perilous situation when one should be especially wary.
Aside from the notion of "incipient moment" or "crucial point" discussed above, the graph for ji by itself indicates "quick-witted(ness); resourceful(ness)" and "machine; device." In combination with other graphs, however, ji can acquire hundreds of secondary meanings. It is absolutely crucial to observe that ji possesses these secondary meanings only in the multisyllabic terms into which it enters. To be specific in the matter under investigation, ji added to hui ("occasion") creates the Mandarin word for "opportunity" (jihui), but by itself ji does not mean "opportunity."
Also, from Straight Dope:
Wu Hung, a Chinese scholar at the University of Chicago, says that originally wei ji didn't even mean crisis. "Ji has a range of meanings, including opportunity but also danger," he says. "When the third-century Chinese began to use the word wei ji, they simply meant danger--a meaning emphasized by both characters."
Not that Kennedy's speech writers cared all that much. Neither did the Swedish city council. I'm sure neither of them expected the Chinese to issue a complaint.