We present an analytical and numerical analysis of neutron multilevel interference phenomena generated when a neutron passes through a series of
N resonant coils operated at the successive conditions $\ensuremath{\hbar}({\ensuremath{\omega}}_{0}+n\ensuremath{\Delta}\ensuremath{\omega})=2{\ensuremath{\mu}}_{n}({B}_{0}+n\ensuremath{\Delta}B)$ with $n=0,1,\dots{},N\ensuremath{-}1$. Each coil produces spin flip with probability $\ensuremath{\rho}$ between 0 and 1; thus the number of waves for the neutron is doubled after each coil, finally giving
2N interfering neutron waves. The phase difference between any pair is a multiple of a time dependent ``phase quantum'' $\ensuremath{\Delta}\ensuremath{\Phi}(t)$. The analysis predicts for each number
N a highly regular pattern for the quantum mechanical probability to find the neutron spin in one specific state as a function of $\ensuremath{\rho}$ and $\ensuremath{\Delta}\ensuremath{\Phi}$. These patterns evolve in time and show revivals after a time
T determined by the step $\ensuremath{\Delta}\ensuremath{\omega}$ according to $T=2\ensuremath{\pi}∕\ensuremath{\Delta}\ensuremath{\omega}$. For some adjustments of the system an analytical solution is obtained. Application of multilevel interference in high-resolution neutron modulated-intensity-by-zero-effort--type spectrometers is discussed.