Is the model failing to converge, as in the 50% posterior HDI clearly does not cover the true parameter 50% of the time or just slow to converge as in the HDI is just so wide all the time?
The second could just mean you don’t have enough data relative to the prior uncertainty to inform the posterior as much as you would like. You can evaluate that by increasing the data size / reducing noise.
Convergence (the first problem) may also be a function of data signal, in that a model will do very poorly for some regimes of data but okay for others.