If by "name" you mean hostname, not really. If you have a domain with multiple nameservers in multiple countries on multiple providers, and your site is similarly globally distributed, at least a few people on the internet will always be able to pull up your site. So at least some of your clients will be able to resolve a domain address from at least some of your nameservers and connect to at least some of your web servers. Geo-IP and Anycast are also really useful here.
edit: It's possible that you could take out an entire TLD and make it impossible to resolve domains on that TLD once all the cached records expire. But that kind of targeted attack would not be possible with a BGP error, unless it was a very specifically crafted BGP error happening over a very long period of time (weeks-months-years depending on the record TTLs).
