NS records exist in two places: in the parent zone (for .com, the registry) as the delegation, and in your own zone. Both sets should match. Resolvers follow the delegation from the root zone down to your name servers.
Changing name servers at the registrar moves the whole zone. All records must exist at the new provider first, otherwise website and mail go down.
Typical problem: someone edits a record at the old DNS provider while the domain has long been delegated elsewhere. The change has no effect at all.
Example
dig +short NS example.com