Time Machine backup error 19 with NAS

Time Machine has been working flawlessly for automatically backing up my MacBook over WiFi for many months. However it has recently needed some help to get started manually—taking a very long time “preparing backup” but finally succeeding. But now it won’t backup even if I start it manually. After spending a long time “preparing backup” it reports “The backup disk image …sparsebundle could not be accessed (error 19).” Time Machine can see the backup disk, and the NAS is mounted and accessible through the finder. Any thoughts?