The problem of state estimation with limited information capacity of the coupling channel for oscillatory multi-input-multi-output nonlinear systems is analyzed for systems in Lurie form (linear part plus nonlinearity depending only on measurable outputs) with first-order coder-decoder. It is shown that for oscillatory systems the upper bound of the limit estimation error is proportional to the maximum rate of the coupling signal and inversely proportional to the information transmission rate (channel capacity). The results are applied to state estimation of a 3rd order nonlinear system: Self-excited mechanical oscillator.