Context. We present a study of the mass accretion rates of pre-main sequence (PMS) stars in the cluster Trumpler 14 (Tr 14) in the Carina Nebula. Using optical multiband photometry we were able to identify 356 PMS stars showing Hα excess emission with equivalent width EW(Hα) > 20 Å. We interpret this observational feature as an indication that these objects are still actively accreting gas from their circumstellar medium. From a comparison of the HR diagram with PMS evolutionary models we derive ages and masses of the PMS stars. We find that most of the PMS objects are younger than 10 Myr with a median age of ~3 Myr. Surprisingly, we also find that ~20% of the mass accreting objects are older than 10 Myr. For each PMS star in Trumpler 14 we determine the mass accretion rate (Ṁacc) and discuss its dependence on mass and age. We finally combine the optical photometry with near-IR observations to build the spectral energy distribution (SED) for each PMS star in Tr 14. The analysis of the SEDs suggests the presence of transitional discs in which a large amount of gas is still present and sustains accretion onto the PMS object at ages older than 10 Myr. Our results, discussed in light of recent recent discoveries with Herschel of transitional discs containing a massive gas component around the relatively old PMS stars TW Hydrae, 49 Ceti, and HD 95086, support a new scenario n which old and evolved debris discs still host a significant amount of gas. Aims: Methods: Results: