In general, no, this is called "Hearsay" evidence which is basically an out of court statement. There are specific incidents, however, that could require someone to testify to statements someone else made. For example, if Alice is testifying against Bob, she is allowed to say "Charlie told me 'Bob shot me' before he died." In this incident, Charlie is unable to be cross-examined (being dead) and thus his statement to Alice can be admissable as Alice is making the accusation against Bob based on the fact that Charlie is indeed dead, and Alice was a witness to Charlie's dying utterance.
She could also make such a statement if it requires to testify against her own interests i.e. Alice says "I broke into Bob's office late at night to rob him and saw 'Kill Charlie' in his day planner on his desk." She would also have to identify the offending day planner and the statement (presuming it still exists), but since she is admitting to committing a crime in and of herself, this is admissible. For a real great lesson to this, check out the movie "My Cousin Vinny" and how the phrase "I shot the Clerk" can mean very different things depending on who says it and why cops will tell you that anything you say "Can and will be used against you in a court of law... ". Because of Hearsay rules, they cannot testify to your profession of innocence, but can testify to statements that make you look guilty, even if you are not guilty.
Often times, if there is a documentation of an event, it could be asked to read a statement made in one piece of evidence (usually the affirmative) that contradicts testimony. This usually comes up in Cross-Examination, where the witness says something that doesn't work with the other side's facts. Alice may be asked to read an affidative of Bob's that contradicts her own statement made in direct, though normally the cross-examining lawyer will enter this into the record by saying a statement of "Please read along while I read out-loud... [conflicting statement]" followed by the question on why the two disagree on a fact in the case. In Law and Order, this is probably cut to Alice reading the statement for drama rather than the more cold read you might find in a court room. Real trial lawyers do not want the witness saying anything other than "Yes" or "No" on cross-examination.