Evidence and theory both suggest that the answer is no, magnets do not have any health effect whatsoever. A nice article that surveys the topic notes problems with studies showing positive effects not being blind.
"Many “controlled” experiments are suspect because it is difficult to blind subjects to the presence of a magnet. An example is a randomised trial of powerful magnetic bracelets for the relief of hip and knee osteoarthritis, which reports a significant decrease in pain because of the bracelets. The patients given real magnets could detect them because the magnets often stuck to keys in pockets. Perhaps subjects with magnetic bracelets subconsciously detected a tiny drag when the bracelets were near ferromagnetic surfaces (which are ubiquitous in modern life), and this distracted or otherwise influenced the perceived pain. Patients with fibromyalgia detected which sleeping pads were magnetic by their mechanical properties, by “comfort with the firmness” and thus unblinded the study. In a sophisticated postural assay, where magnetic soles were found to decrease swaying, the authors admit that the magnetic soles could have differed in stiffness from the controls."
Additonally, when two different tests were properly blinded, both the placebo and magnet groups saw equal improvement.
"The use of a magnet for reducing pain attributed to carpal tunnel syndrome was no more effective than use of the placebo device."
"Application of 1 variety of permanent magnet had no effect on our small group of subjects with chronic low back pain."
Another reason the affect of magnets has been studied is the safety of MRIs. A paper studying this found,
"In the absence of ferromagnetic foreign bodies, there is no replicated scientific study showing a health hazard associated with magnetic field exposure and no evidence for hazards associated with cumulative exposure to these fields. The very high degree of patient safety in strong magnetic fields is attributed to the small value of the magnetic susceptibility of human tissues and to the lack of ferromagnetic components in these tissues."
I know that these claims are slightly different from the one you've heard. But at this point, all evidence suggests that magnets do virtually nothing to people. Someone can always invent a more specific or slightly different claim (maybe it only works on Tuesdays, or a particular brand, etc.) but at this point the science suggests rejecting all claims of bio-active magnets unless they are presented with very compelling evidence.