There are two other possibilities. One is that you have a mixed model (which you likely do since the ACF indicates multiple MA terms). Second, yuo might have seasonality. It is rare from what I have seen to have that many higher order lags - so they might reflect a seasonal factor (although I don't know any that would apply at those intervals). I have rarely seen ACF and PACF that are so close to each other in terms of signficant lags even for the higher order lags. But admitedly I am new to time series too.
Everyone has trouble interpreting ACF and PACF, even the experts.