Groundwater-level and rainfall measurements from 37 borewells in the Visakhapatnam district, Andhra Pradesh, India, from 2002 to 2021 were analyzed using Bayesian Neural Networks (BNNs) to comprehend the predictability of groundwater levels. We found chaotic dynamics in the groundwater and rainfall data, but a dominant trend component was seen in the groundwater-level data from phase plots. Dynamics suggest the presence of self-organized criticality/chaos in the groundwater dynamics over decadal time scales. We used BNN prediction models, (i) nonlinear autoregressive (NAR), (ii) nonlinear input-output (NIO) and (iii) nonlinear autoregressive exogenic Input (NARX), to predict the groundwater levels with rainfall and temperature as exogenic inputs. We noticed similar to 94 to 95% prediction accuracy with the NAR model with optimal inputs and similar to 1% improvement with added exogenic input. Interestingly, the study indicates that (i) the dynamics of the groundwater differ significantly from rainfall and temperature in the region, (ii) the nonlinear autoregressive model based on the self-organized dynamics of groundwater-level changes is robust in providing prediction accuracy up to similar to 95%, and (iii) the dynamics of remaining similar to 5% groundwater-level changes may be due to the presence of randomly varying extreme weather events and man-made/anthropogenic changes.