Sentences such as I came across a letter that was typed by her secretary, can be "reduced" by removing that and the auxiliary verb, yielding I came across a letter typed by her secretary. In this particular case, the by phrase, or some other modifying phrase such as that morning or in red ink appears to be obligatory. You can't simply say, *I came across a letter typed, although you can say I came across a typed letter.
In other cases, though, this seems to be OK; consider All the students arrested were from King's College.
In what cases is a dependent required to make the clause grammatical? Can anyone point me to a discussion of this in a reliable reference grammar, preferably the Cambridge Grammar of the English Language? I see it mentioned there on p. 78, but this particular issue is not discussed.