Can you have someone co-sign for your car purchase if they currently have a co-signer on their car?

They can on in theory as many cars their credit will allow. It will be another debt they are liable for. The bank/institution that would do the approval process may consider the individual 'overextended' if he were to do such, and would not allow him to.

If your intended co-signer needed a co-signer for their own loan then chances are their credit isn't good enough for another lender to accept them as a co-signer on your loan. After all, the bank's purpose for requiring a co-signer is to make certain the loan will be paid. They will go after the co-sogner if the primary borrower defaults on the loan.