What Happened to the Promises to David?
I've been studying Psalms a lot lately. I think the first and last Psalms in each book make a huge difference on how you read each section. My favorite pair is Psalms 89 and 90. I think all of book 3 (in some ways, 1-2 as well) leads up to the problems addressed in Psalms 89 — what happened to the promises to David? Then book 4 answers these questions by taking the reader on a panorama of God's past faithfulness, beginning with Moses in the wilderness in Israel's infancy. "You have been our dwelling place in all generations."
Looking at the book as being compiled from a post-exile perspective (Psalms 137 is proof enough of that), the big question on Israel's mind is "What happened to the promises to David?" Books 1-2 are almost all by or about David, ending with Psalms 72, a song by (or to) Solomon. It's like reviewing Israel's kingship from its beginning (Psalms 2 being about the promises to David), down through it's failure (Psalms 89), then presenting, as Bryant Brailles pointed out, the answers in books 4 and 5.
I also think Psalms 1 and 2 stand out as an introduction to the whole book. In some ways Psalms 3 sets the tone for book 1 more than Psalms 1 does. I have a little harder time seeing a strong transition in Psalms 42-43, but book 2 has that ending "the prayers of David, the son of Jesse are ended," so books 1-2 may be a little more of a single unit, but I'm not sure.
In book 5, the collection of the Egyptian Hallel (113-118), Psalm 119, then the psalms of ascent (120-134) is no accident, I think. The Hallel was recited at Passover (113-114 before, 115-118 after), and reflects on God's faithfulness, even in the face of death. Then Psalms 119 is the monster psalm, all about God's perfect word -- Him keeping His promises! Then, after all that affirmation, the psalms of ascent reflect on going up to Jerusalem (what the people were longing to do in exile), which represents God answering the longing of Psalms 89 with the return. And then, after a three psalm interlude (135-137), a big chunk of David's psalms is back! "The return of the King," if you will.