Can you be sued on a debt that is over 15 years old. A dog was purchased and no contract was signed

You can be sued (anybody can be sued for anything at any time), but it is likely that you could get the case dismissed as barred by limitations.  Although I'm aware of no statute of limitations for breach of contract that runs more than 6 years (I don't know Nevada's statute, but I'm confident it is no more than 6 years), the period runs from the date of the breach, not from the date of the contract.  When was payment due?  Also, limitations periods can be extended under certain circumstances, although you've mentioned nothing in your email that would provide a basis for such an extension.