This should put the reports to rest that the White iPhone 4 is 0.2mm thicker. Phil Schiller is actually right in his tweet.
Soon after the white iPhone 4 was released reports appeared that the new model is 0.2mm thicker. The story raised enough attention that Phil Schiller answered an iPhone owner that the white iPhone 4 is not thicker and that he should not believe all the junk he reads. Now reputable Consumer Reports has taken the matter into their own hands. In their lab they used high-quality calipers and found that both iPhones have the same thickness (0.37 inches). See image above.
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So you do not have to worry that an iPhone 4 case you buy will not fit on your white Phone 4. Why these reports appeared in the first place is not clear. Maybe it is a production issue and some white iPhone 4 are indeed thicker.
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Either way, Consumer Reports uses the opportunity to state that they still recommend getting a case for the iPhone 4 because of the reception issue.