Hopefully you do not have more than one emergency! The emergency fund is restricted to a one-time pay-out regardless of the amount but up to $3,000. The fund is here to help you in your time of need.
If after your emergency the final vet bill ends up being less than $3,000, you will not be credited the remaining balance, and no additional funds will be available to you in case of a new emergency during that year. Your emergency fund will replenish every year upon your sign-up date.