Yes your cousin can file. He'll need to show a substantial change in circumstances to overcome the one year bar to filing. If the country conditions are such that he has a colorable claim of persecution or credible fear of returning, then he should strongly consider filing.
Probably not but there may be alternatives such as Withholding of removal or Convention Against Torture Relief. He should consult an attorney for case assessment.
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There are exceptions to the one year rule. Changes in country conditions and extraordinary circumstances are some, but the application must also be filed within a reasonable time from those events.
He can file, however, he will need to show substantial reason as to why he did not file during his first year. There are two other forms of relief Convention against torture and withholding of removal that do not have the one year rule.
Have a consultation with an attorney that practices immigration law to review his possibilities.
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If conditions in his country changed materially, he can apply for asylum now, but he must do it soon after the conditions change. It is VERY advisable that he use a lawyer experienced with late asylum applications.
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Andrew M. Bramante, Rosner Partners, 216-771-5588. Free telephone consultation. You should always consult with an experienced immigration attorney to make certain that the advice you received is appropriate for your particular immigration case.