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TIJUANA, Mexico — Maggie Alejos arrived here in June from St. Anne, Ill., with her husband, her daughter and a cashier’s check for $13,500, payable to the Regenerative Medicine Institute.

 Rail-thin, with an oxygen tube anchored above her upper lip, Ms. Alejos, a retired Army nurse, has coped with emphysema for a dozen of her 65 years. Once she came close enough to a lung transplant that doctors prepared her for surgery, only to discover that the donor lung was unfit.

At a hospital here, doctors affiliated with the institute extracted about seven ounces of fat from her thighs, hoping to harvest about 130 million stem cells and implant them in her failing lungs.

Across the Internet — where Ms. Alejos learned about the Tijuana institute — adult stem cells are promoted as a cure for everything from sagging skin to severed spinal cords.

On the surface, the claim is plausible. Scientists have discovered that fat, bone marrow and other parts of the body contain stem cells, immature cells that can rejuvenate themselves, at least in the tissue they are...