It's very hard to definitively answer this question, since we know so little of Ilúatar's mind. Personally, I would be inclined to argue "probably not", largely based on this passage from the Silmarillion (emphasis mine):
[I]t is said that after the departure of the Valar there was silence, and for an age Ilúvatar sat alone in thought. Then he spoke and said: 'Behold I love the Earth, which shall be a mansion for the Quendi and the Atani! But the Quendi shall be the fairest of all earthly creatures, and they shall have and shall conceive and bring forth more beauty than all my Children; and they shall have the greater bliss in this world. But to the Atani I will give a new gift.' Therefore he willed that the hearts of Men should seek beyond the world and should find no rest therein; but they should have a virtue to shape their life, amid the powers and chances of the world, beyond the Music of the Ainur, which is as fate to all things else; and of their operation everything should be, in form and deed, completed, and the world fulfilled unto the last and smallest.
The Silmarillion III Quenta Silmarillion Chapter 1: "Of the Beginning of Days"
From this passage, it appears as though Ilúvatar is giving each of the Children different gifts, not necessarily giving either a better lot. Of course, the nature of us limited incarnate creatures being what it is, both sides think the other got the better deal. The Elves grow weary of living forever amid the suffering of the world, while Men fear the unknown fate that awaits them, and look upon the Elves as rather the more impressive of the two races.
We also see Tolkien himself thinking along these lines; for example, in Letter 181:
In this mythological world the Elves and Men are in their incarnate forms kindred, but in the relation of their 'spirits' to the world in time represent different 'experiments', each of which has its own natural trend, and weakness. The Elves represent, as it were, the artistic, aesthetic, and purely scientific aspects of the Humane nature raised to a higher level than is actually seen in Men.
But the Elvish weakness is in these terms naturally to regret the past, and to become unwilling to face change: as if a man were to hate a very long book still going on, and wished to settle down in a favourite chapter.
The Letters of J.R.R. Tolkien 181: To Michael Straight (draft). January/February 1956
I think it's ultimately important to realize that Elves and Men have different roles in this mythical universe, and each race does something different for the other. The Elves were, broadly speaking, tasked with removing the threat of incarnate Evil (in the form of Morgoth and, to a lesser degree, Sauron1).
On the other hand, Ilúatar has some plan for Men, but nobody really knows what that is; at least some of the Elves believe that Men will be the ones to heal the world of the damage inflicted by Morgoth in the Music of the Ainur:
'This then, I propound, was the errand of Men, not the followers, but the heirs and fulfillers of all: to heal the Marring of Arda, already foreshadowed before their devising; and to do more, as agents of the magnificence of Eru: to enlarge the Music and surpass the Vision of the World!'
History of Middle-earth X Morgoth's Ring Part 4: "Athrabeth Finrod ah Andreth"
But this is only a guess of Finrod's; all we know for sure is that both Elves and Men will participate in the Second Music, following Melkor's ultimate destruction.
1 Though Sauron isn't that simple; Sauron's threat was mostly the result of the "fall" of the Elves, and he probably wouldn't have been defeated without their assistance; though the ultimate responsibility was that of Men.