Even if Dumbledore would be able to come back, he wouldn't. This is one of the primary points of the books.
Although I agree with the other answers that the questioner seems to have misunderstood the Hallows, I want to point out that even if Dumbledore would have been able to come back, he wouldn't.
The primary theme of Harry Potter is about overcoming the fear of death:
Death is an extremely important theme throughout all seven books. I would say possibly the most important theme. (Accio-quote)
Voldemort's main personality trait, and the one that defines him the most, is his fear of death.
Voldemort's fear is death, ignominious death. I mean, he regards death itself as ignominious. He thinks that it's a shameful human weakness, as you know. His worst fear is death, but how would a boggart show that? I'm not too sure. I did think about that because I knew you were going to ask me that.
“There is nothing worse than death, Dumbledore!” snarled
This is also his greatest weakness:
“Your failure to understand that there are things much worse than death has always been your greatest weakness—”
In fact all the bad guys primary traits seem to be based on fear of death
Only innocent lives, Peter!”
“You don’t understand!” whined Pettigrew. “He would have
killed me, Sirius!”
However, the defining characteristic of the good guys is their willingness to embrace death. This is evident from the first book:
"To one as young as you, I'm sure it seems incredible, but to Nicolas
and Perenelle, it really is like going to bed after a very, very long
day. After all, to the well-organized mind, death is but the next great
adventure. You know, the Stone was really not such a wonderful thing. As
much money and life as you could want! The two things most human beings
would choose above all -- the trouble is, humans do have a knack of
choosing precisely those things that are worst for them."
This is expressed in many other places in the books as well, but here's one of my favorite quotes:
“THEN YOU SHOULD HAVE DIED!” roared Black. “DIED
RATHER THAN BETRAY YOUR FRIENDS, AS WE WOULD
HAVE DONE FOR YOU!”
And Snape becomes good only when he seems to embrace death:
“DON’T!” bellowed Snape. “Gone . . . dead . . .”
“Is this remorse, Severus?”
“I wish . . . I wish I were dead. . . .”
Dumbledore, although never clearly seeming to have fear of death, definitely viewed himself as selfish and wanted to overcome death, and believes that his selfishness may caused him caused him to avoid sacrificing himself to Ariana (in his view):
“I know how you are feeling, Harry,” said Dumbledore very
That in my mind, is one of the most important overlooked quotes in the book (and
one that will be of utmost importance in the Fantastic Beasts movies.)
That quote comes right after Harry blames himself for Sirius's death:
It was his fault Sirius had died; it was all his fault.
This means that, Like Harry, Dumbledore as well blamed himself for someone's death, and in the later books, we find out it was Arianna.
“Don’t hurt them,
don’t hurt them, please, please, it’s my fault, hurt me instead . . .”
“He thought he was back there with you and Grindelwald, I
know he did,” said Harry, remembering Dumbledore whimpering,
pleading. “He thought he was watching Grindelwald hurting you
and Ariana. . . . It was torture to him, if you’d seen him then, you
wouldn’t say he was free.”
All those closest to Albus — and I count myself
one of that lucky number — agree that Ariana’s
death, and Albus’s feeling of personal responsibility
for it (though, of course, he was guiltless), left their
mark upon him forevermore.
And we know we why he felt guilty - because he ignored his sister while he tried to overcome death:
“And at the heart of our schemes, the Deathly Hallows! How
they fascinated him, how they fascinated both of us! The unbeatable
wand, the weapon that would lead us to power! The Resurrection
Stone — to him, though I pretended not to know it, it meant an
army of Inferi! To me, I confess, it meant the return of my parents,
and the lifting of all responsibility from my shoulders.
“And the Cloak . . . somehow, we never discussed the Cloak much,
Harry. Both of us could conceal ourselves well enough without the
Cloak, the true magic of which, of course, is that it can be used to
protect and shield others as well as its owner. I thought that, if we
ever found it, it might be useful in hiding Ariana, but our interest
in the Cloak was mainly that it completed the trio, for the legend
said that the man who united all three objects would then be truly
master of death, which we took to mean ‘invincible.’
“Invincible masters of death, Grindelwald and Dumbledore!
Two months of insanity, of cruel dreams, and neglect of the only
two members of my family left to me.
“And then . . . you know what happened. Reality returned in
the form of my rough, unlettered, and infinitely more admirable
brother. I did not want to hear the truths he shouted at me. I did
not want to hear that I could not set forth to seek Hallows with a
fragile and unstable sister in tow.
“The argument became a fight. Grindelwald lost control. That
which I had always sensed in him, though I pretended not to, now
sprang into terrible being. And Ariana . . . after all my mother’s care
and caution . . . lay dead upon the floor.”
Harry on the other hand, is always portrayed as selfless:
“Do not misunderstand me,” he said, and pain crossed the face
so that he looked ancient again. “I loved them. I loved my parents,
I loved my brother and my sister, but I was selfish, Harry, more
selfish than you, who are a remarkably selfless person, could possibly
Thus, only Harry, who is selfless and pure and has zero fear of death, was able to choose selflessly to live.
But Dumbledore, even if his possession of the Hallows allowed him to choose life, would not have, because he knew inherently that his choice would include some selfishness and desire to live, and thus was not purely selfless. He would have feared making this choice, as he knew himself to be undeserving. Thus he went to the opposite extreme and chose death - by his own hand - through a hallow, to overcome his selfishness.
This is also implied in the tale of Beatle the Bard, that only the brother who made the selfless choice (the cloak) was allowed to survie death, or more so, embrace him like a brother.