In fact, if language has happened, we always know what has been deleted. As in the examples above, deletions are very common in oral language. Here is a bit of dialog. In written English deleting either the primary NP or the VP will generally create trouble. Any other sentence that lacks an expressed noun phrase or verb phrase is a fragment.
Fragments are very bad.
The boy with the long yellow hair and the unsightly blemishes underneath his chin. Waiting quietly under a huge cedar tree, thumb in mouth, blanket in hand. It is hardest to pick out fragments in context, for there the deletions are usually immediately recoverable. Four hundred weeping kindergarten kids and only one ice-cream truck. The beach was a madhouse.