It used to be the case (and probably still is, thought a quick google says it may vary by store) that Lowes and Home Depot would honor each other's coupons.
Lowes, and I assume Home Depot too, assign unique numbers to each 10% coupon so that no coupon can be used twice. As an aside Lowes had a problem recently with coupon arriving invalid because people had figured out the pattern they used for making the coupons and used them online before Lowes sent out the coupons.
Home Depot cannot check to see if a Lowes coupon has been use and the same is true in reverse.
What has likely happened is the coupon has been used online, so it wasn't surrendered, and is no longer valid at the issuer's store. But, if competitors will price match based on the coupon it can still be used for that purpose.