Can you have multiple DNS registrars?
Solution 1:
You are not understanding the difference between a DNS service provider and a registrar, probably because most registrars will also act as DNS service providers if you want/pay them to.
A registrar takes your $10/year or whatever and makes sure that your domain ends up in the giant list of all domains. So if you pay GoDaddy to register "example.com", GoDaddy takes your $10 and sends $8 of it to Verisign (the operator of the .com top-level domain) and asks Verisign to please tell the world that example.com exists, that you are the "owner", and that a small group of DNS servers will be responsible for telling the rest of the world how to communicate with example.com.
That's it, the registrar's job is done, other than updating the WHOIS contact database if you move, change DNS service, or transfer the domain.
Your DNS provider(s) run the machines that are listed with the .com domain as the authoritative sources of information about your domain.
You can be your own DNS provider. You can pay someone else to be your DNS provider; many registars do this for free as part of their yearly charge, but it's not technially part of a registrar's job.
So you could have register.com register your domain - let's say they will provide you with 2 DNS hosts as part of their package. You set up your domain information with them to say that you're going to have four authoritative servers for your domain - register.com's 2 nameservers, and 2 from dnsmadeeasy.com. You would then set up all of your A and CNAME records on register.com's systems, and set up their systems to allow a "zone transfer" to the nameservers for dnsmadeeasy.com You would set up dnsmadeeasy.com to provide "secondary" or "slave" DNS service, pulling their information from the master copy at register.com.
In this scenario, you have one registrar - register.com. You have two DNS service providers - register.com and dnsmadeeasy.com. Each DNS server would act as a "failover" for the others - although I suspect it's more likely that your host or your network connection will fail.
Solution 2:
You can have as many DNS providers as you like, but the domain name itself can only be registered with a single organization.
Solution 3:
You can only have one registrar per domain name.
That is the company that will have the contract with the relevant TLD registry for the provision of your domain name.
You could have multiple independent domain hosters for your domain, but you had better be damned sure they always have the same information in their copy of the domain, or things will break.
Usual practise is to have one primary hoster, and then use secondary hosting too. The secondary hoster would need to be able to "zone transfer" the zone from the primary hoster.