If Lucius Malfoy is correct, and by the evidence of Broderick Bode it seems that he is,
[T]he only people who are permitted to retrieve a prophecy from the Department of Mysteries, Potter, are those about whom it was made, as the Dark Lord discovered when he attempted to use others to steal it for him.
Harry Potter and the Order of the Phoenix - p.693 - Bloomsbury - Chapter 35, Beyond the Veil
(See also Adamant's answer)
'Avery told me Bode would be able to remove it.'
'Bode could never have taken it, Master ... Bode would have known he could not ... undoubtedly, that is why he fought so hard against Malfoy's Imperius Curse ...'
Harry Potter and the Order of the Phoenix - p.516 - Bloomsbury - Chapter 26, Seen and Unforeseen
I think that there is no doubt that Neville could not have picked it up.
The crucial quote, for me, follows on from the quote Adamant provides:
'Then - it might not be me?' said Harry.
'I am afraid,' said Dumbledore slowly, looking as though every word cost him a great effort, 'that there is no doubt that it is you.'
'But you said - Neville was born at the end of July, too - and his mum and dad -'
'You are forgetting the next part of the prophecy, the final identifying feature of the boy who could vanquish Voldemort ... Voldemort himself would mark him as his equal. And so he did, Harry. He chose you, not Neville. He gave you the scar that has proved both blessing and curse.'
'But he might have chosen wrong!' said Harry. 'He might have marked the wrong person!'
'He chose the boy he thought most likely to be a danger to him.' said Dumbledore. 'And notice this, Harry: he chose, not the pure-blood (which, according to his creed, is the only kind of wizard worth being or knowing) but the half-blood, like himself. He saw himself in you before he had ever seen you, and in marking you with that scar, he did not kill you, as he intended, but gave you powers, and a future, which have fitted you to escape him not once, but four times so far - something that neither your parents, nor Neville's parents, ever achieved.'
Harry Potter and the Order of the Phoenix - p.742 - Bloomsbury - Chapter 37, The Lost Prophecy
The Prophecy is quite explicit.1
the Dark Lord will mark him as his equal,
This was Harry, not Neville.
but he will have power the Dark Lord knows not
Well Neville appears to have had that power too, the power of love, to a great extent. But it was certainly Harry who walked into that Hall of Prophecy with his mother's charm protecting him. And Harry was certainly shown to be somewhat unique in the depth of his love, and in its endurance:
'You are protetcted [from the desire to become one of Voldemort's followers], in short, by your ability to love!' said Dumbledore loudly. 'The only protection that can work against the lure of power like Voldemort's! In spite of the temptation you have endured, all the suffering, you remain pure of heart, just as pure as you were at the age of eleven, when you stared into a mirror that reflected your heart's desire, and it showed you only the way to thwart Lord Voldemort, and not immortality or riches. Harry, have you any idea how few wizards could have seen what you saw in that mirror?
Harry Potter and the Half-Blood Prince - p.478 - Bloomsbury - Chapter 23, Horcruxes
Would Neville have shown such love? Would Neville's mother have died for him and given him an unbreakable protection? Possibly. But it doesn't matter. Because she didn't, did she? It didn't happen to Neville, did it? The Prophecy tells of someone who will have a power that the Dark Lord knows not, and a part of that seems to be his mother's charm, and Neville doesn't have that, whether he could have had, or would have had is immaterial, he does not have.
And in any case, the Dark Lord will mark him as his equal. The Prophecy foretells Voldemort picking a boy, and that boy will be given the power to defeat him. Voldemort picks a boy and gives that boy the power to defeat him. That boy is not Neville.
1Harry Potter and the Order of the Phoenix - p.741 - Bloomsbury - Chapter 37, The Lost Prophecy