I guess my first question is where have you often heard this asserted?

It's not implausible that it would be common to have much of the psalter memorised, simply based on early church practices. We do know that the psalter comprised the bulk of liturgical singing, that in ascetic contexts there would be chanting/singing of multiple psalms at different times of the day, and that even in Late Antiquity we are discussing a society that was much more orally orientated than we are. All these factors would suggest that a bishop would do a lot better on a memory test of passages from the psalms than, say, a modern one would do.

But without seeing where this has been asserted, I'm loath to start going down the reference trail to find documentary support for it.