Changing Name Servers

On a domain record at DirectNIC, there is room for up to 6 name servers.
I am in the process of relocating several domains from one server to
another. The present name server entries are ns1.oldserver.com and
ns2.oldserver.com. If I add ns1.newserver.com and ns2.newserver.com as
the third and fourth entries now, when I make the change this weekend,
if I take down the site on oldserver, will traffic automatically find
its way to newserver? I would then, of course, update the domain record
to exclude the entries for oldserver.

Obviously I'm trying to make the transition as smooth as possible to
avoid the problems with DNS changes not propagating simultaneously all
over the world.

I've never seen this done, so there are probably good reasons for me not
to do it. I thought I would ask.