Can one leave out data from research because it is not significant? I've encountered this sentence while reading an article on sciencemag.org.

In the end, responses from just 7600 researchers in 12 countries were included because the remaining data were not considered statistically significant. 

Is this a proper way to do research?  To leave out results because they were not considered statistically significant?
 A: No.
I suspect the reporter meant to say that the other individuals were left out because the surveys were incomplete or internally inconsistent.
A: In the report cited in whuber's comment, it says on page 104 [pg 114 in the pdf]:

The survey succeeded in activating the participation of approximately 8,900 doctoral candidates from more than 30 countries...

Then, spanning pages 104-105, it says:

While conducting data cleaning procedures, the Eurodoc survey experts' team decided to run a power test analysis.  Based on the assumption of fully completed questionnaires which will result in a multi normal distribution, a power test for estimation of the confidence interval was used.  This was done to test the accuracy of the data.  It was decided to accept maximum a 6% error-level at a 95% confidence interval.  A loss of 16% of the sampling size resulted in a sample of 12 participating countries with 7,600 participants.

So it's not really clear exactly why the 16% loss in the sample, but the assumption of incomplete responses is likely correct.  (And you can see why the reporter was confused.)
A: No, but reporters can use technical jargon completely nonsensically.
A: That sentence does not actually make sense and is clearly in error.
Data cannot be statistically significant or insignificant. Only relationships between data, the product of statistical tests, can be spoken about in these terms.
If the question is: Can we drop data from our analyses because the inclusion of that data means we cannot reject the null hypothesis? The answer is — obviously, I hope! — no. The message you've cited is a news report, not a scientific paper. Had it been a paper that was reviewed, it never would have gotten in.
Probably, data was not included because there are substantive reasons to not include those data. Probably, as others have suggested, the excluded data was incomplete or collected using different or incomparable methods.
