In 2009 the US Federal Aviation Administration (FAA) introduced a slow growth approach for certifying composite and adhesively-bonded structures. This approach requires that delamination or disbond is slow, stable and predictable under cyclic-fatigue loads. The present paper addresses the challenge of developing a methodology capable of enabling this approach to certification to be implemented. To this end we have examined the growth of disbonds from small naturally-occurring material discontinuities in adhesively-bonded structures. It is shown that, for the examples studied, the disbond growth histories can be accurately computed using a form of the Hartman and Schijve variant of the NASGRO crackgrowth equation. It is also shown that the scatter in the disbond growth histories can be captured by allowing for small changes in the fracture-mechanics threshold term. These findings suggest that the Hartman-Schijve crack-growth equation has the potential to address the 'slow growth' approach to certifying composite/bonded structures and bonded repairs outlined in the US FAA Airworthiness Advisory Circular No: 20-107B.