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 (ω0 +nΔω) =2 μn (B0 +nΔB) with n=0,1,...,N-1. Each coil produces spin flip with probability ρ 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" ΔΦ (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 ρ and ΔΦ. These patterns evolve in time and show revivals after a time T determined by the step Δω according to T=2π/Δω. 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.