Secondary neutrons generated during ion beam radiotherapy present a concern due to the potential dose deposition beyond the treatment volume, thereby elevating the risk of inducing secondary tumours. These neutrons can possess energies comparable to those of the primary ions, reaching magnitudes of several hundred MeV, posing a challenge for neutron detectors. Fluorescent Nuclear Track Detectors (FNTDs) are promising detectors for high-energy neutron dosimetry given their capability to detect particles with a low linear energy transfer. In this work, the sensitivity of FNTDs to neutron energies reaching 20 MeV was analysed by experiments and Monte Carlo (MC) simulations, quantifying the recoil proton yield of FNTDs combined with polyethylene (PE) converters of different thicknesses. The FNTDs were read out using a dedicated FNTD reader, demonstrating a reasonable uncertainty by analysing a detector area of 0.1 mm². Investigations of different converter thicknesses reveal optimal detector sensitivity between 0.5 mm to 1.0 mm for a ²⁴¹AmBe source, yielding a maximum sensitivity of (22.7±3.4) tracks mSv^-1 mm^-2. Similar converter-FNTD configurations were assessed through MC simulations using FLUKA, yielding a correlation between detector sensitivity and converter thickness. Furthermore, an enhanced detector sensitivity for neutron energies up to 20 MeV was found for the PE converter thickness of 4.0 mm. The MC simulations can be used to optimise FNTD detector configurations for measuring higher neutron energies by maximising the recoil proton yield.