On the basis of the tight-binding model, we have studied the energy spectra and persistent currents (PCs) in one-dimensional k-component Fibonacci (KCF) mesoscopic rings threaded by a magnetic flux. The KCF structures, which contain k basic units, can be periodic (if k = 1), quasiperiodic (if 1 < k < 6), and intermediate cases between quasiperiodicity and disorder (if k 6). It is shown that the flux-dependent eigenenergies form 'band' structure in the KCF rings. The subbands possess the hierarchical characteristic with self-similarity if 1 < k < 6, while if k 6, there is no obvious self-similarity in the subbands. In fact, the energy spectra ultimately determine the behaviour of the PCs in the mesoscopic KCF rings. On one hand, the PC depends on the total energy bandwidth: the narrower the bandwidth, the smaller the PC. On the other hand, the parity effect of electrons is dissimilar in different KCF rings. As k increases, there is less likelihood of observing a dramatic change in currents of several orders of magnitude when one electron is added to or removed from the KCF rings. If k is large enough, the current behaviour may approach some features of disordered systems.