Probably not, since most organisms (as opposed to cell cultures) don't die because of the Hayflick limit. If humans were killed by the Hayflick limit, their lifespans would be at least 150 years (according to the latest estimates.)
But in any case, the Hayflick limit is not an absolute constraint: cells can be made to divide forever, blasting the Hayflick limit out of the water.
"That seems to be a good general framework for thinking about cellular senescence in cells grown in the laboratory," agrees Cech. But he predicts that the mechanisms that dictate cellular life span will inevitably turn out to be more complex. For example, he notes that other researchers have found that other cells with an added catalytic subunit gene will not divide beyond their normal limit unless they are also lacking p53, a well-known tumor suppressor gene that is mutated in many human cancers.