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In general, no, this wouldn't be a problem and the banks would cash the checks, but, many banks at their own discretion will not cash checks dated longer than six months ago.
They are allowed to refuse to cash checks dated more than six months ago under the Uniform Commercial Code
I'm afraid I cannot give you a more definitive answer than that. I would suggest that you try to deposit the checks, and if the bank refuses to honor them then contact the court or your previous tenant to get replacement checks.
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