…with a lot of things up them. So far up the sleeves, in fact, that they’re close to the chest.
Requisite heavily-laboured metaphors dispensed with, I’ll (attempt to) get to the point.
Despite Dumbledore’s claims at the end of Order of the Phoenix that he would tell Harry “everything”, we find he is still imparting information to Harry in Half-Blood Prince. Harry quite rightly calls him out on this, and his response is that he has conveyed all the facts, and the information he is imparting now is conjecture.
Apart from this simply reeking of palliation*, it has the potential to undermine Harry’s trust. How can he believe any assurances Dumbledore makes in the future?
But this mitigation aside, Dumbledore is just flat-out lying. There are lots of other things he knows that would be of use to Harry in his future quest to defeat Voldemort. He could tell him about Professor Snape’s motivations**. That the Sword of Gryffindor is now basilisk-powered for improved pest control. That Draco has taken the Dark Mark, Dumbledore and Snape know about his mission, and are monitoring him. That his cloak is The OMGWTFBBQ Invisibility Cloak of Hallowed Deathlyness. But he doesn’t.
So, why not? Well, besides the fact that book seven would have been much shorter, there does seem a strong character-based reason: Dumbledore is old, learned, wise, and a teacher. This has certain side-effects.
1) In order to teach effectively, it helps to understand the students’ perspective. Unfortunately, once you are experienced enough with any particular subject, there are details that you start to take for granted. It therefore becomes very hard to put yourself in the shoes of someone without that experience. One of Dumbledore’s biggest mistakes is forgetting that not everyone is as clever as he is (despite his frequent false modesty on the topic).
2) Teacher’s often try to lead their students to work things out for themselves, giving them the tools then offering hints/nudges as appropriate. Dumbledore shows exactly this tendency when sharing memories of Voldemort’s past with Harry. He also keeps certain information (e.g. how he injured his hand) tantilisingly out of reach, again likely a habit from encouraging students to strive.
3) It is easy to get caught in the role of teacher and ignore the possibility of learning anything from your students (which is missing out on a lot, in my experience). We have no indication that Dumbledore speaks/understands parseltongue, yet he does not ask Harry to translate for him. It’s possible that someone else has already done so, but given a) the scarcity of parselmouths, and b) Dumbledore’s reluctance to share information, this seems unlikely.
So, in some ways it’s kind of inevitable. There’s even a grim portent as Dumbledore remarks that
“…being—forgive me—rather cleverer than most men, my mistakes tend to be correspondingly huger.”***
* It’s similar to the “from a certain point of view” excuse used by Obi-Wan Kenobi. Technically correct, but not the way any reasonable person would interpret the statement.
** He may not know where Snape’s ultimate loyalties lie, but he has fairly strong evidence. He certainly does know what has happened in the past, and some details (not necessarily all) could have been helpful to Snape and Harry working together. Then again, maybe he wanted them at obvious loggerheads so as to preserve Snape’s cover and… oh dear, I’ve gone cross-eyed.