On U.S. Treasury checks, if the check is more than a year old the check is void. It's not stale dated, it's void; it will not be paid. The teller should circle the date and hand the check back to the customer with a suggestion that he or she contact the issuing agency and obtain a new check.
On state checks, you would need to contact the Treasurer of the state that issued the check for their individual perspective. There are no uniform rules, but I would expect any check which they felt to be stale dated in a period less than 6 months would contain a legend to that effect.
On business or personal checks, the paying bank could pay the check or it could return it - you would have no way of knowing what decision they might make. If your teller noted the stale date I would expect that he or she would simply circle the date, explain the problem to the customer and hand it back, instructing the customer to request a replacement check. Alternatively, the bank could accept the item for deposit and impose an exception hold based on reasonable cause to doubt collectibility. Finally, the bank could send the check as a collection item.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.