Some of the less typical pictorial vignettes are veneer but thicker than the cheaper stuff which is more common, less valuable and can look like paint peeling away. Certain vignettes / scenes will be far more mass produced and so the veneer tends to be thinner. Sometimes the same vignette / scene will have been used on different pieces.
There are other decorative designs which are often classed as Tunbridge Ware but have little in common with the place. However, these designs do retain a certain ' quality ' and individuality.
Fine quality marquetry and parquetry boxes have inlaid pieces like jigsaw puzzles, not paint thin veneer. If thin then the design and work has to be superbly done..that's when pieces can fetch top dollar.
Yours looks like a nice piece of wood was used though.
That's what i meant when i said it's easy to refer to any inlaid box as Tunbridge Ware when strictly speaking there are reasons for it
Hope i've cleared that up a bit but is after midnight here !