Customers may observe that their PaaS role instances and IaaS virtual machines are only issued one DNS server IP address by DHCP. This does not mean that name resolution in Azure has a single point of failure however.
The Azure DNS infrastructure is highly redundant. The IP address that is exposed to the customer virtual machine is a virtual IP address in the Azure platform. That virtual IP address maps to a cluster of DNS servers in the same region that are behind a load balanced IP so a failure of any particular server is not a concern. In the event a DNS server cluster in the region fails, the virtual IP address exposed to customers will fail over to a DNS server cluster in a nearby region. The only impact of such a failure to customers will be a slight increase in latency.