Why DID Dumbledore have Jame's cloak?

[Bmat]

From JKR's site:

NAQ
... which means, 'never asked question'.

Why did Dumbledore have James' invisibility cloak at the time of James' death, given that Dumbledore could make himself invisible without a cloak?

Prior to posting this I had a quick look on-line, and realised that some fans have been speculating about this question. However, nobody has ever asked me about it, and they really should have done. Just to allay the fears of the justifiably suspicious, this isn't what we in the know call 'a Mark Evans situation.'* There IS a significant - even crucial - answer.

Has this been answered already? If not, anybody have speculation?

[aldan]

Well, MY speculation is that the Dumbledore that was slain wasn't the REAL Dumbledore, but instead was someone else either transformed or under an illusion. The cloak, perhaps, being a magic item, wouldn't transform or be covered by the illusion... or something.

[Merle]

That mystery hasn't been answered yet, Bmat. It is to be answered in Deathly Hallows.

Dumbledore heard the prophecy, and would know that there were two boys who it could refer to: Harry or Neville. Until V marked Harry, it was still up in the air. So one would assume that V would go after both families. We know that the Potters hid their residence. Did the Longbottoms also do this? And why, if James knew V was hunting him, would he give away his invisibility cloak? I have no speculation on why, just those few background thoughts.