It is discovered theoretically that significant confinement resonances in an nl photoionization of a multielectron atom A encaged in carbon fullerenes, A@C60, may re-appear and be strong at photon energies far exceeding the nl ionization threshold, as a general phenomenon. The reasons for this phenomenon are unraveled. The Ne 2p photoionization of the endohedral anion Ne@C 5− 60 in the photon energy region of about a thousand eV above the 2p threshold is chosen as case study. PACS numbers: 32.80.Fb,31.15.V Endohedral fullerenes A@C 60 , where an atom A is confined (encaged) inside a hollow carbon cage C 60 , are modern frontline targets of research in chemistry and physics. This is in view of their novelty in basic science and importance to various applied sciences and technologies. In particular, many efforts have been undertaken to unravel trends in the response of A@C 60 confined atoms A, referred to as A@ for brevity, to external perturbations, such as the incoming photoionizing radiation (see Refs. [1,2,3,4,5] and references therein) and fast chargedparticles [6]. One of outstanding inherent features of the corresponding spectra is the presence of resonances, termed confinement resonances [1,2,4]. New aspects of confinement resonances are the subject of this paper.Much of the current understanding of the nature and origin of confinement resonances in spectra of confined atoms A@C 60 is based on modeling the C 60 cage by a short-range attractive spherical potential V c (r) of inner radius r 0 = 5.8 a.u., depth U 0 = −8.2 eV, and either the zero thickness, i.e., V c (r) = −U 0 δ(r − r 0 ) [4], or finite thickness ∆ = 1.9 a.u. [1,2,6,7,8] (and references therein):The formation of the A@C 60 system is completed by placing the neutral atom A at the center of the cage. For small sized, compact atoms there is no charge transfer to the cage, so that the confined atom A@ retains the structure of the free atom A. In the framework of such modeling, confinement resonances in partial nl ionization cross sections of the A@ atom occur due to the interference of the ejected photoelectron waves emerging directly from the confined atom, and those scattered off the confining C 60 cage. According to the thus accumulated database of calculated data, confinement resonances in a partial nl ionization cross section of a A@C n system have been known to rapidly vanish with increasing energy of the outgoing photoelectron, ceasing to exist at only some tens eV above the nl threshold, not to mention thousands eV above the threshold. This is in line with a theory of scattering of particles off a potential well/barrier. Indeed, starting from a sufficiently high energy of the outgoing electron, the corresponding coefficient of reflection off a finite potential well/barrier decreases with increasing energy of the electron. As a result, the interference effect between the outgoing and scattered electron waves becomes weaker, with increasing energy of the electron, and so are the associated confinement resonances. For the case of A@C 60 p...