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Is it true that one of the family cord blood banks stores “twice as many stem cells” as the others?

This claim is mis-leading. The exact wording of their advertising is "up to twice as many" stem cells compared to other cord blood banks. Quite frankly, if it really were true that their processing method is so efficient, then the other cord blood laboratories would have rushed to copy it.

The bank in question is a marketing company that does not operate their own laboratory; they sub-contract laboratory services to an independent company. Moreover, they have switched laboratories several times recently. It is very hard to believe that their methods are twice as good as the proprietary services that established banks have developed with in-house research.