Only if the DNS work. If you are doing DNS from in-house as well and someone cuts off your internet service, your email will start to hard bounce.
This isn't necessarily true. A name error (3) is only meaningful if it comes from an authoritative name server. If the DNS server is offline, the requesting DNS server should also return server failure (2) if the designated name server does not respond. An SMTP server should never give up on delivery if the DNS lookup simply fails. If I try to send mail to 'extra.co.nz', which exists but doesn't have MX, A or AAAA records, my server will quite happily keep retrying delivery until the message is expired form the queue. That would be similar to the DNS simply being offline for a while, or a temporary misconfiguration of the domain. If I send to a domain that doesn't exist at all, I will get an immediate notification and the mail will be removed from the queue.