I would add to what has been said that it's their bad fortune, but it's also the values that society held up to the Romantic movement. Poor people were "interchangable," without the dignity of the upper classes. But the source of human dignity isn't what you have, and Romantic movement soon to come woud support the dignity of the common man that "Elegy" does:
Let not Ambition mock their useful toil,
Their homely joys, and destiny obscure;
Nor Grandeur hear with a disdainful smile
The short and simple annals of the Poor.
The boast of heraldry, the pomp of power,
And all that beauty, all that wealth e'er gave,
Awaits alike th' inevitable hour:-
The paths of glory lead but to the grave.
The seeming glories of the rich are really not all that important because they will all be lost in death. If society were to see each individual as what they are, then greatness would come from being, not possessions.
In Thomas Gray's "Elegy Written in a Country Churchyard," poverty is said to have stopped the villagers from achieving the kind of "greatness" achieved by, say, Cromwell or senators. Gray writes:
But Knowledge to their eyes her ample page
Rich with the spoils of time did ne'er unroll:
Chill Penury repressed their noble rage,
And froze the genial current of the soul. (49-52)
One might say that lack of knowledge and poverty prevented them, but due to the use of the colon, I interpret these lines to mean that knowledge did not unroll because of their poverty. Others may disagree.
Concerning lines 85-92, these lines deal with the human need to be remembered after death. One can go back as far as the Greek epics to demonstrate how central this is to the human mind. A poem fulfills this need, if it's exceptional and a little lucky, anyway, by immortalizing its subject.
In this case, the churchyard and the villagers buried in it are immortalized by Gray's poem.
The real reason why the people buried in this country churchyard were unable to achieve greatness is the fact that they were born unlucky. It is merely their "lot" to have been born in the wrong sort of situation.
Because they were born unlucky, they were never able to get enough education to allow them to advance in the world.
But Knowledge to their eyes her ample page,
Rich with the spoils of time, did ne'er unroll;
Because they were born unlucky, they were poor and were prevented from advancing. Their drive was suppressed by poverty.
Chill Penury repress'd their noble rage,
And froze the genial current of the soul.
The people buried in the courtyard simply had the bad luck to be born poor and were therefore unable to get the education they needed.
Gray's "Elegy" is one of the major lyrics of the eighteenth century, and one of the representative poems of the "graveyard school" of poetry, a major theme of which was the need for living a sensible, good life in view of the inevitability of death. As a biographical note, you might point out that Gray himself was the only one of his parents' twelve children to grow to adulthood. A concern with death and how to take life is therefore not an unexpected aspect of his art.
The setting of day of the poem is twilight. The cattle are heading back to the barn to be milked, the farmer is returning from the fields, the sun is setting, and the curfew bell is ringing from the church tower. For much of the poem, the speaker seems to be addressing no one in particular, but in line 37 he does address “yeproud,” and in line 93 he seems to speak to a person who is buried in the country churchyard.
The people buried in the church graveyard are humble, rural folk. Yet the speaker asserts that they are not contemptible because of their simplicity; instead he emphasizes their “useful toil” and “homely joys,” pointing out that death is the great leveler, and that “the paths of glory lead but to the grave” (one of the more famous lines of English literature). Some of those buried here might have made great rulers, musicians, defenders of human rights, or poets. But the speaker balances the missed opportunities for good that are buried here by pointing out that the people never had the chance to do evil either. In short, the churchyard is the occasion of reflection on the need for goodness and piety, and the inevitability of death is cause for people to live their lives to their fullest potential.