Attention is given to a population of 110 stars in the NGC 6611 cluster of the Eagle Nebula that have prominent near-infrared (NIR) excess and optical colours typical of pre-main sequence (PMS) stars older than 8 Myr. At least half of those for which spectroscopy exists have a Halpha emission line profile revealing active accretion. In principle, the V-I colours of all these stars would be consistent with those of young PMS objects (< 1 Myr) whose radiation is heavily obscured by a circumstellar disc seen at high inclination and in small part scattered towards the observer by the back side of the disc. However, using theoretical models it is shown here that objects of this type can only account for a few percent of this population. In fact, the spatial distribution of these objects, their X-ray luminosities, their optical brightness, their positions in the colour-magnitude diagram and the weak Li absorption lines of the stars studied spectroscopically suggest that most of them are at least 8 times older than the ~1 Myr-old PMS stars already known in this cluster and could be as old as ~30 Myr. This is the largest homogeneous sample to date of Galactic PMS stars considerably older than 8 Myr that are still actively accreting from a circumstellar disc and it allows us to set a lower limit of 7% to the disc frequency at ~16 Myr in NGC 6611. These values imply a characteristic exponential lifetime of ~6 Myr for disc dissipation.
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